of 45 /45

# Review Mastery Guide

Tags:

• #### print format

Embed Size (px)

DESCRIPTION

Holt Geometry Review for Mastery solution guide.

### Text of Review Mastery Guide

- .3._JK4. __ EF * + % &Name each figure using words and symbols.5. CD6. 43 line CD or ___ CD ray ST or ___ ST 7.Name the plane in two different ways.8. 78 ,-.

plane LMN; plane Q segment WX ;_WX Ppoint PCopyright by Holt, Rinehart and Winston.2Holt GeometryAll rights reserved.1-1LESSONTerm Meaning Modelcollinear points that lie on the same line&'(F and G are collinear.F, G, and H are noncollinear.noncollinear points that do not lie on the same linecoplanar points or lines that lie in the same plane:789W, X, and Y are coplanar.W, X, Y, and Z are noncoplanar.noncoplanar points or lines that do not lie in the same planeFigures that intersect share a common set of points. In the first model above, __ FH intersects __ FG at point F. In the second model, __ XZ intersects plane WXY at point X.Use the figure for Exercises 914. Name each of the following.*#0!\$+"

9.three collinear points10.three noncollinear points Possible answers: A, P, and B; Sample answer: A, P, and D 11.four coplanar points12.four noncoplanar points Sample answer: C, P, B, and D Sample answer: J, D, P, and B 13.two lines that intersect ___ CD14.the intersection of __ JK and plane R ___ AB and __ JK point DReview for MasteryUnderstanding Points, Lines, and Planes continuedC, P, and D; J, D, and KCopyright by Holt, Rinehart and Winston.3Holt GeometryAll rights reserved.LESSON LESSON1-2Review for MasteryMeasuring and Constructing SegmentsThe distance between any two points is the length of the segment that connects them. centimeters (cm)0 1 2 3 4 5 6 7% & ' ( *The distance between E and J is EJ, the length of_EJ . To find the distance, subtract the numbers corresponding to the points and then take the absolute value.EJ 7 1 6 6 cm Use the figure above to find each length.1.EG2.EF3.FH 4 cm 1.5 cm 3 cm001 12 022 1 X On_PR , Q is between P and R. If PR 16, we can find QR. PQ + QR PR 9 x 16x 7QR 74. * + ,

Y5. ! " #

ZFind JK.2Find BC.66. 3 4 6

NN7. 7 8 9

A AFind SV.41Find XY.218. \$ % & X9. 3 4 5

YYFind DF.135Find ST.22Copyright by Holt, Rinehart and Winston.4Holt GeometryAll rights reserved.LESSON1-2Review for MasteryMeasuring and Constructing Segments continuedSegments are congruent if their lengths are equal.AB BCThe length of_AB equals the length of_BC . _AB _BC _AB is congruent to_BC .Copying a SegmentMethod Stepssketch using estimation Estimate the length of the segment. Sketch a segment that is about the same length.draw with a ruler Use a ruler to measure the length of the segment. Use the ruler to draw a segment having the same length.construct with a compass and straightedgeDraw a line and mark a point on it. Open the compass to the length of the original segment. Mark off a segment on your line at the same length.Refer to triangle ABC above for Exercises 10 and 11. 10.Sketch_LM that is congruent to_AC .11.Use a ruler to draw_XY that is congruentto_BC . 12.Use a compass to construct_ST that is congruent to_JK . *+ 3 4The midpoint of a segment separates the segment into two congruent segments. In the figure, P is the midpoint of_NQ .

.X X0 1 13. _PQ is congruent to _NP or_PN . 14.What is the value of x? 4 15.Find NP, PQ, and NQ. 12, 12, 24 ! #"

:There are four ways to name this angle.YUse the vertex.XYZ or ZYXUse the vertex and a point on each side.2Use the number.Name each angle in three ways.1.0 21

2.(+*Q, PQR, 1 J, HJK, KJH3.Name three different angles in the figure. \$!#"ABD, ABC, DBCAngle acute right obtuse straightModelAAAAPossible Measures0 a 90 a 90 90 a 180 a 180Classify each angle as acute, right, obtuse, or straight.4.NMP .1-0, obtuse5.QMN right6.PMQ acuteThe vertex is Y.The sides are __ YX and __ YZ .Copyright by Holt, Rinehart and Winston.6Holt GeometryAll rights reserved.LESSON1-3Review for MasteryMeasuring and Constructing Angles continuedYou can use a protractor to find the measure of an angle. '%\$ &1OO8O11O7O12OGO18O5O 14O 4O 15O 8O1GO 2O17O1O8O1OO7O11O GO12O5O18O4O14O8O15O2O1GO1O17OOOUse the figure above to find the measure of each angle.7. DEG8. GEF55 125The measure of XVU can be found by adding. 6875

mXVU mXVW mWVU 48 48 96Angles are congruent if their measures are equal. In the figure, XVW WVU because the angles have equal measures. ___ VW is an angle bisector of XVU because it divides XVU into two congruent angles.Find each angle measure.\$%

!"#&9.mCFB if AFC is a straight angle.10.mEFA if the angle is congruent to DFE.102 51 11.mEFC if DFC AFB.12.mCFG if __ FG is an angle bisector of CFB.129 51DEG is acute.GEF is obtuse.Copyright by Holt, Rinehart and Winston.7Holt GeometryAll rights reserved.LESSONReview for MasteryPairs of Angles 1-4Angle PairsAdjacent Angles Linear Pairs Vertical Angleshave the same vertex and share a common sideadjacent angles whose noncommon sides are opposite raysnonadjacent angles formed by two intersecting lines

1 and 2 are adjacent. 3 and 4 are adjacent and form a linear pair. 5 and 6 are vertical angles.Tell whether 7 and 8 in each figure are only adjacent, are adjacent and form a linear pair, or are not adjacent.1.

2.

5.1 and 4 not adjacent6.2 and 3 adjacent and form a linear pairName each of the following.7.a pair of vertical angles Possible answers:1 and 6, 2 and 5

8.a linear pair Possible answers: 1 and 2; 1 and 5; 5 and 6; 6 and 29.an angle adjacent to 4 3Copyright by Holt, Rinehart and Winston.8Holt GeometryAll rights reserved.LESSON1-4Review for MasteryPairs of Angles continuedAngle PairsComplementary Angles Supplementary Anglessum of angle measures is 90 sum of angle measures is 180

m1 m2 90In each pair, 1 and 2 are complementary. m3 m4 180In each pair, 3 and 4 are supplementary.Tell whether each pair of labeled angles is complementary, supplementary, or neither. 10. 11.

complementary neitherFind the measure of each of the following angles. 12.complement of S 34 3

13.supplement of S 124 14.complement of R 68 2

15.supplement of R 158 16.LMN and UVW are complementary. Find the measure of each angle if mLMN (3x 5) and mUVW 2x. mLMN 56; mUVW 34Copyright by Holt, Rinehart and Winston.3Holt GeometryAll rights reserved.Copyright by Holt, Rinehart and Winston.9Holt GeometryAll rights reserved.LESSONReview for MasteryUsing Formulas in Geometry 1-5The perimeter of a figure is the sum of the lengths of the sides. The area is the number of square units enclosed by the figure.Figure Rectangle SquareModelW W

SSSSPerimeter P 2 2w or 2( w) P 4sArea A w A s 2Find the perimeter and area of each figure.1.rectangle with 4 ft, w 1 ft2.square with s 8 mm 10 ft; 4 ft2 32 mm; 64 mm23. CM4.ININX X 28 cm; 49 cm2 (24 2x) in.; 12x in2The perimeter of a triangle is the sum of its side lengths. The base and height are used to find the area.HACB BCHA PerimeterAreaP = a + b + cA =1 __ 2 bh orbh ___ 2

M 1 7 _____ 2 ,2 6 _____ 2 = M 8 __ 2 ,8 __ 2 = M(4, 4)Find the coordinates of the midpoint of each segment.1. 6XY06 3"(4, 5) ! (2, 5)2. 3XY03 334 (1, 4)3(3, 2) (1, 5) (1, 1)3. _QR with endpoints Q(0, 5) and R(6, 7) (3, 6)4. _JK with endpoints J(1, 4) and K(9, 3) (5, 0.5)Suppose M(3, 1) is the midpoint of_CD and C has coordinates (1, 4). You can use the Midpoint Formula to find the coordinates of D.M (3, 1) M x1 x2 ______ 2,y1 y2 ______ 2

x-coordinate of Dy-coordinate of D3 x1 x2 ______ 2 Set the coordinates equal.1 y1 y2 ______ 2 3 1 x2 ______ 2Replace (x1, y1) with (1, 4).1 4 y2 ______ 2 6 1 x2Multiply both sides by 2.2 4 y25 x2Subtract to solve for x2 and y2.6 y2The coordinates of D are (5, 6).5. M(3, 2) is the midpoint of_RS , and R has coordinates (6, 0). What are the coordinates of S? (12, 4)6. M(7, 1) is the midpoint of_WX , and X has coordinates (1, 5). What are the coordinates of W? (15, 3)M is the midpoint of_HG .Copyright by Holt, Rinehart and Winston.12Holt GeometryAll rights reserved.LESSON1-6Review for MasteryMidpoint and Distance in the Coordinate Plane continuedThe Distance Formula can be used to find the distance d7XY70! (1, 2)"(7, 6)Dbetween points A and B in the coordinate plane.d (x2 x1)2 (y2 y1)2 (7 1 )2 (6 2)2(x1, y1) (1, 2); (x2, y2) (7, 6)62 42Subtract. 36 16Square 6 and 4. 52Add. 7.2Use a calculator.Use the Distance Formula to find the length of each segment or the distance between each pair of points. Round to the nearest tenth.7. _QR with endpoints Q(2, 4) and R(3, 9)8. _EF with endpoints E(8, 1) and F(1, 1)7.1 units 9 units9. T(8, 3) and U(5, 5)10. N(4, 2) and P(7, 1)8.5 units 11.4 unitsYou can also use the Pythagorean Theorem to find distances in the coordinate plane. Find the distance between J and K.c2 a2 b2Pythagorean Theorem

XY

* + CBA 52 62a 5 units and b 6 units 25 36Square 5 and 6. 61Add.c 61or about 7.8Take the square root.Use the Pythagorean Theorem to find the distance, to the nearest tenth, between each pair of points. 11.6XY06 3:(4, 5)9 (0, 1)12.XY

2 123 31and S(0, 0). After a transformation, the image of the figure has vertices at Q(1, 4), R(4, 3), and S(0, 0). The transformation is a rotation.A translation can be described using a rule such as (x, y) (x 4, y 1).Preimage Apply Rule ImageR(3, 5) R(3 4, 5 1) R(7, 4)S(0, 1) S(0 4, 1 1) S(4, 0)T(2, 1) T(2 4, 1 1) T(6, 2)Draw each figure and its image. Then identify the transformation.5.Triangle HJK has vertices at H(3, 1),

XY *(++*(J(3, 4), and K(0, 0). After a transformation, the image of the figure has vertices at H(1, 3), J(1, 2), and K(4, 2). translation6.Triangle CDE has vertices at C(4, 6),

XY

\$ #% %# \$D(1, 6), and E(2, 1). After a transformation, the image of the figure has vertices at C(4, 6), D(1, 6), and E(2, 1). reflectionFind the coordinates for each image after the given translation.7.preimage: XYZ at X(6, 1), Y(4, 0), Z(1, 3)rule: (x, y) (x 2, y 5) X(4, 6), Y(6, 5), Z(3, 8)8.preimage: FGH at F(9, 8), G(6, 1), H(2, 4)rule: (x, y) (x 3, y 1) F(6, 9), G(9, 2), H(5, 5)9.preimage: BCD at B(0, 2), C(7, 1), D(1, 5)rule: (x, y) (x 7, y 1) B(7, 1), C(0, 0), D(8, 4)Copyright by Holt, Rinehart and Winston.15Holt GeometryAll rights reserved.LESSONReview for MasteryUsing Inductive Reasoning to Make Conjectures 2-1When you make a general rule or conclusion based on a pattern, you are using inductive reasoning. A conclusion based on a pattern is called a conjecture.Pattern Conjecture Next Two Items8, 3, 2, 7, . . . Each term is 5 more than the previous term.7 5 1212 5 1745The measure of each angle is half the measure of the previous angle.22.511.25Find the next item in each pattern.1. 1 __ 4 ,1 __ 2 ,3 __ 4 , 1, . . .2.100, 81, 64, 49, . . . 1 1 __ 4 363. 3 6 104.

perpendicularlines that form 90 anglessymbol: skew lines that do not lie in the same plane and do not intersectParallel planes are planes that do not intersect. For example, the top and bottom of a cube represent parallel planes.Use the figure for Exercises 13. Identify each of the following.1.a pair of parallel lines JGH

X5You can write and solve an inequality for x.WU WT _WT is the shortest segment.x 1 8Substitute x 1 for WU and 8 for WT. 1 1Subtract 1 from both sides of the equality.x 7Use the figure for Exercises 1 and 2.1.Name the shortest segment from point K to __ LN.

_KM 2.Write and solve an inequality for x. , -+

X. x 5 14; x 9Use the figure for Exercises 3 and 4.3.Name the shortest segment from point Q to ___ GH.

_QH 4.Write and solve an inequality for x. x 2 9; x 11 '( 1

XLine b is the perpendicular bisector of_RS .The shortest segment from W to __ SU is_WT.Copyright by Holt, Rinehart and Winston.36Holt GeometryAll rights reserved.LESSONReview for MasteryPerpendicular Lines continued 3-4You can use the following theorems about perpendicular lines in your proofs.Theorem ExampleIf two intersecting lines form a linear pair of congruent angles, then the lines are perpendicular.Symbols:2 intersecting lines form lin. pair of lines .AB1 21 and 2 form a linear pair and 1 2, so a b.Perpendicular Transversal TheoremIn a plane, if a transversal is perpendicular to one of two parallel lines, then it is perpendicular to the other line.Symbols: Transv. Thm.DCHh c and c d, so h d.If two coplanar lines are perpendicular to the same line, then the two lines are parallel to each other.Symbols:2 lines to same line 2 lines .KJ

X'(* &Y

X2 345Y

9. __ FG and __ HJ for F(1, 2), G(3, 4), 10. __ RS and __ TU for R(2, 3), S(3, 3), H(2, 3), and J(4, 1) T(3, 1), and U(3, 1) perpendicular neitherCopyright by Holt, Rinehart and Winston.39Holt GeometryAll rights reserved.LESSONslope y-intercept slopeReview for MasteryLines in the Coordinate Plane 3-6Slope-Intercept Form Point-Slope Formy mx by 4x 7y y1 m(x x1)point on the line:y 2 1 __ 3 (x 5)(x1, y1) (5, 2)Write the equation of the line through (0, 1) and (2, 7) in slope-intercept form.Step 1: Find the slope.m y2 y1 ______x2 x1 Formula for slope7 1 _____ 2 0 6 __ 2 3Step 2: Find the y-intercept.y mx bSlope-intercept form1 3(0) bSubstitute 3 for m, 0 for x, and 1 for y.1 bSimplify.Step 3: Write the equation.y mx bSlope-intercept formy 3x 1Substitute 3 for m and 1 for b.Write the equation of each line in the given form.1.the line through (4, 2) and (8, 5) in 2.the line through (4, 6) with slope1 __ 2 slope-intercept formin point-slope formy 3 __ 4 x 1 y 6 1 __ 2 (x 4)3.the line through (5, 1) with slope 2 4.the line with x-intercept 5 and in point-slope formy-intercept 3 in slope-intercept formy 1 2(x 5)y 3 __ 5 x 35.the line through (8, 0) with slope 3 __ 4 6.the line through (1, 7) and (6, 7) in slope-intercept formin point-slope formy 3 __ 4 x 6y 7 0Copyright by Holt, Rinehart and Winston.40Holt GeometryAll rights reserved.LESSON3-6Review for MasteryLines in the Coordinate Plane continuedYou can graph a line from its equation.Consider the equation y 2 __ 3 x 2.y-intercept 2 slope 2 __ 3

XY

First plot the y-intercept (0, 2). Use rise 2 and run 3 to find another point. Draw the line containing the two points.Parallel Lines Intersecting Lines Coinciding Lines

XY

same slopedifferent y-intercepts XY

different slopes XY

same slopesame y-interceptGraph each line.

X

Y

X

Y

XY

7. y x 28. y 1 __ 3 x 39. y 2 1 __ 4 (x 1)Determine whether the lines are parallel, intersect, or coincide. 10. y 2x 511. y 1 __ 3 x 4y 2x 1x 3y 12parallel coincide 12. y 5x 213.5y 2x 1x 4y 8y 2 __ 5 x 3intersect parallelrun: go left 3 unitsrise: go up 2 unitsy 1 __ 3 x 2y 1 __ 3 xy 1 __ 2 x 2y 2x 1y 2 __ 3 x 12x 3y 3Copyright by Holt, Rinehart and Winston.11Holt GeometryAll rights reserved.Copyright by Holt, Rinehart and Winston.41Holt GeometryAll rights reserved.LESSONReview for MasteryClassifying Triangles 4-1You can classify triangles by their angle measures. An equiangular triangle, for example, is a triangle with three congruent angles.Examples of three other triangle classifications are shown in the table.Acute Triangle Right Triangle Obtuse Triangleall acute angles one right angle one obtuse angleYou can use angle measures to classify JML at right.JLM and JLK form a linear pair, so they are supplementary.mJLM mJLK 180Def. of supp. mJLM 120 180SubstitutionmJLM 60Subtract.Since all the angles in JLM are congruent, JLM is an equiangular triangle.Classify each triangle by its angle measures.1.

2.

3.

right obtuse acuteUse the figure to classify each triangle by its angle measures.4. DFGright5. DEGacute6. EFG obtuse!"#N!"# is equiangular.!# "6060 60

*- +,

\$' &%

JKL is obtuse so JLK is an obtuse triangle.Copyright by Holt, Rinehart and Winston.42Holt GeometryAll rights reserved.LESSONReview for MasteryClassifying Triangles continued 4-1You can also classify triangles by their side lengths.Equilateral Triangle Isosceles Triangle Scalene Triangleall sides congruentat least two sides congruent no sides congruentYou can use triangle classification to find the side lengths of a triangle.Step 1Find the value of x.QR RSDef. of segs.4x 3x 5Substitutionx 5Simplify.Step 2Use substitution to find the length of a side.4x 4(5)Substitute 5 for x. 20Simplify.Each side length of QRS is 20.Classify each triangle by its side lengths.7.EGF isosceles8.DEF scalene9.DFG isoscelesFind the side lengths of each triangle. 10. XX11. XXX 9; 9; 9 7; 7; 4

X X21 3\$' &%

Copyright by Holt, Rinehart and Winston.43Holt GeometryAll rights reserved.LESSONReview for MasteryAngle Relationships in Triangles 4-2mC 90 39 51According to the Triangle Sum Theorem, the sum of the angle

*, +measures of a triangle is 180. mJ mK mL 62 73 45 180The corollary below follows directly from the Triangle Sum Theorem.Corollary ExampleThe acute angles of a right triangle are complementary.

% \$#mC mE 90Use the figure for Exercises 1 and 2.1.Find mABC.

!\$#"472.Find mCAD.38Use RST for Exercises 3 and 4.3.What is the value of x? (7X13)(4X9)(2X2)24 3144.What is the measure of each angle?mR 85; mS 30; mT 65What is the measure of each angle?

. ,- "#! X5765. L6. C7. W49 39.8 (90 x)Copyright by Holt, Rinehart and Winston.44Holt GeometryAll rights reserved.LESSONAn exterior angle of a triangle is formed by

one side of the triangle and the extension of an adjacent side.1 and 2 are the remote interior angles of 4 because they are not adjacent to 4.Exterior Angle TheoremThe measure of an exterior angle of a

triangle is equal to the sum of the measures of its remote interior angles.Third Angles TheoremIf two angles of one triangle are congruent

to two angles of another triangle, then the third pair of angles are congruent.Find each angle measure.

XY

XY

1.a square with side lengths of 6 units2.a right triangle with leg lengths of 3 units and 4 units Possible answer on graph above. Possible answer on graph above.

XY

XY

3.a triangle with a base of 8 units and 4.a rectangle with a length of 6 units and a height of 2 unitsa width of 3 units Possible answer on graph above. Possible answer on graph above.4-7Review for MasteryIntroduction to Coordinate ProofCopyright by Holt, Rinehart and Winston.54Holt GeometryAll rights reserved.LESSONYou can prove that a statement about a figure is true without knowing the side lengths. To do this, assign variables as the coordinates of the vertices.

XYDCPosition each figure in the coordinate plane and give the coordinates of each vertex.

XYTS

XYK KKK5.a right triangle with leg lengths s and t 6.a square with side lengths k Possible answer on graph above. Possible answer on graph above.

XYWW

XYHB7.a rectangle with leg lengths and w 8.a triangle with base b and height h Possible answer on graph above. Possible answer on graph above.9.Describe how you could use the formulas for midpoint and slope to prove the following.Given: HJK, R is the midpoint of_HJ , S is the midpoint of_JK .Prove:_RS _HK Possible answer: Use the midpoint formula to find the coordinates of the midpoints R and S. Then use the coordinates and the formula for slope to find the slopes of_RS and_HK.4-7Review for MasteryIntroduction to Coordinate Proof continueda right triangle with leg lengths c and dCopyright by Holt, Rinehart and Winston.55Holt GeometryAll rights reserved.LESSONTheorem ExamplesIsosceles Triangle TheoremIf two sides of a triangle are congruent, then the angles opposite the sides are congruent.4 32If_RT _RS , then T S.Converse of Isosceles Triangle TheoremIf two angles of a triangle are congruent, then the sides opposite those angles are congruent.. -,If N M, then_LN _LM .You can use these theorems to find angle measures in isosceles triangles.Find mE in DEF.mD mEIsosc. Thm. \$ %&X X 5x (3x + 14)Substitute the given values.2x 14Subtract 3x from both sides.x 7Divide both sides by 2.Thus mE 3(7) 14 35.Find each angle measure. ! #"

2 01

"Given: is the perpendicular bisector of_FG .Conclusion: AF AGThe Converse of the Perpendicular Bisector Theorem is also true. If a point is equidistant from the endpoints of a segment, then it is on the perpendicular bisector of the segment.You can write an equation for the perpendicular bisector of a segment. Consider the segment with endpoints Q (5, 6) and R (1, 2).Step 1 Find the midpoint of_QR .Step 2 Find the slope of the bisector of_QR . x1 x2 ______ 2,y1 y2 ______ 2 5 1 _______ 2 ,6 2 _____ 2

y2 y1 ______ x2 x1 2 6 ________ 1 (5) Slope of_QR (2, 4) 2 __ 3 So the slope of the bisector of_QR is3 __ 2 .Step 3 Use the point-slope form to write an equation.y y1 m (x x1)Point-slope formy 4 3 __ 2 (x 2)Slope 3 __ 2 ; line passes through (2, 4), the midpoint of_QR .Find each measure.2M4 31614 6 \$T" !2.54# *+3X15X3( ,1. RT 162. AB 53. HJ 7Write an equation in point-slope form for the perpendicular bisector of the segment with the given endpoints.4. A (6, 3), B (0, 5)5. W (2, 7), X (4, 3)y 1 3 __ 4 (x 3) y 5 3 __ 2 (x 1)Each point on is equidistant from points F and G.Copyright by Holt, Rinehart and Winston.58Holt GeometryAll rights reserved.LESSONReview for MasteryPerpendicular and Angle Bisectors continued 5-1Theorem ExampleAngle Bisector TheoremIf a point is on the bisector of an angle, then it is equidistant from the sides of the angle.

-0,.Given: ___ MP is the angle bisector of LMN.Conclusion: LP NPConverse of the Angle Bisector TheoremIf a point in the interior of an angle is equidistant from the sides of the angle, then it is on the bisector of the angle. -0,.Given: LP NPConclusion: ___ MP is the angle bisector of LMN.Find each measure. ' &%(

2431

8:7 9X X6.EH 7.mQRS 8.mWXZ 23 52 35Use the figure for Exercises 911. ( *+,9.Given that __ JL bisects HJK and LK 11.4, find LH. 11.4 10.Given that LH 26, LK 26, and mHJK 122, find mLJK. 61 11.Given that LH LK, mHJL (3y 19), and mLJK (4y 5), find the value of y. 14Point P is equidistant from sides __ ML and ___ MN .LMP NMPCopyright by Holt, Rinehart and Winston.59Holt GeometryAll rights reserved.LESSON4. 13-20Theorem ExampleCircumcenter TheoremThe circumcenter of a triangle is equidistant from the vertices of the triangle.Given:_MR ,_MS , and_MT are4. 13-20the perpendicular bisectors of NPQ.Conclusion: MN MP MQIf a triangle on a coordinate plane has two sides that lie along the axes, you can easily find the circumcenter. Find the equations for the perpendicular bisectors of those two sides. The intersection of their graphs is the circumcenter. _HD,_JD, and_KD are the perpendicular bisectors of EFG. &+% '( *\$

Find each length.1. DG2. EK 19 173. FJ4. DE 15 19Find the circumcenter of each triangle.5.XY#\$ /

6.XY,(0, 5)-(8, 0) /(0, 0)34(2, 3) ( 4, 2.5)Review for MasteryBisectors of Triangles 5-2The point of intersection of_MR ,_MS , and_MT is called the circumcenter of NPQ.Perpendicular bisectors _MR ,_MS , and_MT are concurrent because they intersect at one point.Copyright by Holt, Rinehart and Winston.60Holt GeometryAll rights reserved.LESSON' *(!Theorem ExampleIncenter TheoremThe incenter of a triangle is equidistant from the sides of the triangle.Given:_AG ,_AH , and_AJ are'\$#"*(!the angle bisectors of GHJ.Conclusion: AB AC AD _WM and_WP are angle bisectors of MNP, and WK 21. +- 0.7

Find mWPN and the distance from W to_MN and_NP .mNMP 2mNMWDef. of bisectormNMP 2(32) 64Substitute.mNMP mN mNPM 180 Sum Thm.64 72 mNPM 180Substitute.mNPM 44Subtract 136 from each side.mWPN 1 __ 2 mNPMDef. of bisectormWPN 1 __ 2 (44) 22Substitute.The distance from W to_MN and_NP is 21 by the Incenter Theorem. _PC and_PD are angle bisectors of CDE. Find each measure. #\$10%

7.the distance from P to_CE8.mPDE 9 34 _KX and_KZ are angle bisectors of XYZ. Find each measure. : 8+9

Step 2 Find equations of the lines containing two altitudes.The altitude from A to_BC is the horizontal line y 3.The slope of__AC 0 3 ________3 (3) 1 __2 , so the slope of the altitudefrom B to_AC is 2. The altitude must pass through B(3, 7). y y1 m(x x1) Point-slope form y 7 2(x 3)Substitute 2 for m and the coordinates of B (3, 7) for (x1, y1).y 2x 1Simplify.Step 3Solving the system of equations y 3 and y 2x 1, you find that the coordinates of the orthocenter are (1, 3).Triangle FGH has coordinates F (3, 1), G (2, 6), and H (4, 1).9.Find an equation of the line containing theXY'( &

altitude from G to_FH .x 2 10.Find an equation of the line containing the altitude from H to_FG . y x 5 11.Solve the system of equations from Exercises 9 and 10 to find the coordinates of the orthocenter.(2, 3)Find the orthocenter of the triangle with the given vertices. 12.N (1, 0), P (1, 8), Q (5, 0)13.R (1, 4), S (5, 2), T (1, 6) (1, 1) (3, 2)Review for MasteryMedians and Altitudes of Triangles continued 5-3The point of intersection of the altitudes is called the orthocenter of JKL. _JD ,_KE , and_LC are altitudes of a triangle. They are perpendicular segments that join a vertex and the line containing the side opposite the vertex.Copyright by Holt, Rinehart and Winston.63Holt GeometryAll rights reserved.LESSONA midsegment of a triangle joins the midpoints of two sides of the triangle. Every triangle has three midsegments.3# %2\$Use the figure for Exercises 14._AB is a midsegment of RST.1.What is the slope of midsegment_AB and the slopeXY3(2, 3)!(1, 0)4(6, 1)"(3, 2)2(0, 3)223 0of side_ST ?1; 12.What can you conclude about_AB and_ST ?Since the slopes are the same,_AB _ST .3.Find AB and ST.AB 22 , ST 4 2 4.Compare the lengths of_AB and_ST . AB 1 __ 2 ST or ST 2ABUse MNP for Exercises 57.5. _UV is a midsegment of MNP. Find theXY.(4, 5)50(2, 1)6-(4, 7)43 3 0coordinates of U and V. U (1, 3), V (3, 2)6.Show that_UV _MN . The slope of_UV 1 __ 4 and the slope of _MN 1 __ 4 . Since the slopes are the same,_UV _MN .7.Show that UV 1 __ 2 MN. UV 17and MN 217 . Since 171 __ 2 (2 17 ), UV 1 __ 2 MN.5-4Review for MasteryThe Triangle Midsegment Theorem _RS is a midsegment of CDE.R is the midpoint of_CD . S is the midpoint of_CE .Copyright by Holt, Rinehart and Winston.64Holt GeometryAll rights reserved.LESSONTheorem ExampleTriangle Midsegment TheoremA midsegment of a triangle is parallel to a side of the triangle, and its length is half the length of that side.1,.0-Given:_PQ is a midsegment of LMN.Conclusion:_PQ _LN , PQ 1 __ 2 LNYou can use the Triangle Midsegment Theorem to"(!# +*

find various measures in ABC. HJ 1 __ 2 AC Midsegment Thm. HJ 1 __ 2 (12)Substitute 12 for AC. HJ 6Simplify. JK 1 __ 2 AB Midsegment Thm. _HJ ||_ACMidsegment Thm.4 1 __ 2 ABSubstitute 4 for JK.mBCA mBJHCorr. Thm.8 ABSimplify.mBCA 35Substitute 35 for mBJH.Find each measure. (* '68 7

8.VX 239.HJ 54 10.mVXJ 92 11.XJ 27Find each measure.3%4#2\$

( *+

5.Write the angles in order from smallest 6.Write the sides in order from shortestto largest.to longest.V, S, T _JK ,_KH ,_HJ Theorem ExampleTriangle Inequality TheoremThe sum of any two side lengths of a triangle is greater than the third side length. ACB a b cb c ac a bTell whether a triangle can have sides with the given lengths. Explain.7.3, 5, 88.11, 15, 21No; 3 5 8, which is notYes; the sum of each pair of greater than the length of thelengths is greater than the length third side. of the third side.5-5Review for MasteryIndirect Proof and Inequalities in One Triangle continued _WY is the longest side.X is the largest angle.Copyright by Holt, Rinehart and Winston.67Holt GeometryAll rights reserved.LESSONTheorem ExampleHinge TheoremIf two sides of one triangle are congruent to two sides of another triangle and the included angles are not congruent, then the included angle that is larger has the longer third side across from it.+ - '(,*If K is larger than G, then side_LM is longer than side_HJ .The Converse of the Hinge Theorem is also true. In the example above, if side_LM is longer than side_HJ , then you can conclude that K is larger than G. You can use both of these theorems to compare various measures of triangles.Compare NR and PQ in the figure at right. 012

3PN QRPR PRmNPR mQRPSince two sides are congruent and NPR is smaller than QRP, the side across from it is shorter than the side across from QRP.So NR PQ by the Hinge Theorem.Compare the given measures.389764

('&-,+

1.TV and XY2.mG and mL TV XY mG mL \$ !"#

('&%

3.AB and AD 4.mFHE and mHFGAB AD mFHE mHFGReview for MasteryInequalities in Two Triangles 5-6Copyright by Holt, Rinehart and Winston.68Holt GeometryAll rights reserved.LESSONYou can use the Hinge Theorem and its converse to find a range of values in triangles.Use MNP and QRS to find the range of values for x..1230-

X

Step 1 Compare the side lengths in the triangles.NM SRNP SQmN mSSince two sides of MNP are congruent to two sides of QRS and mN mS, then MP QR by the Hinge Theorem.MP QRHinge Thm.3x 6 24Substitute the given values.3x 30Add 6 to each side.x 10Divide each side by 3.Step 2 Check that the measures are possible for a triangle.Since_MP is in a triangle, its length must be greater than 0.MP 0Def. of 3x 6 0Substitute 3x 6 for MP.x 2Simplify.Step 3 Combine the inequalities.A range of values for x is 2 x 10.Find a range of values for x.5.

X

6.2623(3X9)542 x 7 3 x 217. 14 10(2X6)1088.

X

3 x 57 0.6 x 75-6Review for MasteryInequalities in Two Triangles continuedCopyright by Holt, Rinehart and Winston.18Holt GeometryAll rights reserved.Copyright by Holt, Rinehart and Winston.69Holt GeometryAll rights reserved.LESSONThe Pythagorean Theorem states that the following relationship exists among the lengths of the legs, a and b, and the length of the hypotenuse, c, of any right triangle.ACBa 2 b 2 c 2Use the Pythagorean Theorem to find the value of x in each triangle.

X

XXa 2 b 2 c 2Pythagorean Theorema 2 b 2 c 2x 2 62 92Substitute.x 2 42 (x 2)2x 2 36 81Take the squares.x 2 16 x 2 4x 4x 2 45Simplify.4x 12x 45x 3x 3 5 Find the value of x. Give your answer in simplest radical form.1.

X2.

Xx 12 x 29 3.

X4.

XXx 39x 405-7Review for MasteryThe Pythagorean TheoremTake the positive square root and simplify.Copyright by Holt, Rinehart and Winston.70Holt GeometryAll rights reserved.LESSONA Pythagorean triple is a set of three nonzero whole numbers a, b, and c that satisfy the equation a 2 b 2 c 2.You can use the following theorem to classify triangles by their angles if you know their side lengths. Always use the length of the longest side for c.Pythagorean Inequalities Theorem!"#CABIf c 2 a 2 b 2, then ABC is obtuse.!"#CABIf c 2 a 2 + b 2, then ABC is acute.Consider the measures 2, 5, and 6. They can be the side lengths of a triangle since 2 5 6, 2 6 5, and 5 6 2. If you substitute the values into c 2 a 2 b 2, you get 36 29. Since c 2 a 2 b 2, a triangle with side lengths 2, 5, and 6 must be obtuse.Find the missing side length. Tell whether the side lengths form a Pythagorean triple. Explain.5.

6.

10; yes; the three side lengths are95 ; no;95is not a whole nonzero whole numbers that number.satisfy a2 b2 c 2.Tell whether the measures can be the side lengths of a triangle. If so, classify the triangle as acute, obtuse, or right.7.4, 7, 9 8.10, 13, 169.8, 8, 11 yes; obtuse yes; acute yes; acute 10.9, 12, 15 11.5, 14, 2012.4.5, 6, 10.2 yes; right no yes; obtuse5-7Review for MasteryThe Pythagorean Theorem continuedPythagorean TriplesNot Pythagorean Triples3, 4, 5,5, 12, 132, 3, 46, 9, 117 mC 90 mC 90Copyright by Holt, Rinehart and Winston.71Holt GeometryAll rights reserved.LESSONTheorem Example45-45-90 Triangle TheoremIn a 45-45-90 triangle, both legs are congruent and the length of the hypotenuse is2times the length of a leg.

qiqi In a 45-45-90 triangle, if a legXX

Xqilength is x, then the hypotenuse length is x 2 .Use the 45-45-90 Triangle Theorem to find the value of x in EFG.Every isosceles right triangle is a 45-45-90 triangle. TriangleX& '%X

EFG is a 45-45-90 triangle with a hypotenuse of length 10.10 x 2Hypotenuse is 2times the length of a leg. 10 ___ 2 x 2 ____ 2 Divide both sides by 2 .52 xRationalize the denominator.Find the value of x. Give your answers in simplest radical form.1. X

2. X

x 17 2x 22 2 3.XX

4. X

qix 4 2x 25Review for MasteryApplying Special Right Triangles 5-8Copyright by Holt, Rinehart and Winston.72Holt GeometryAll rights reserved.LESSON5-8Theorem Examples30-60-90 Triangle TheoremIn a 30-60-90 triangle, the length of the hypotenuse is 2 multiplied by the length of the shorter leg, and the longer leg is 3 multiplied by the length of the shorter leg.

qi qi

In a 30-60-90 triangle, if the shorter leg

XqiXXlength is x, then the hypotenuse length is 2x and the longer leg length is x.Use the 30-60-90 Triangle Theorem to find the values

XY*( +of x and y in HJK.12 x 3Longer leg shorter leg multiplied by 3 . 12 ___ 3 xDivide both sides by 3 . 4 3 xRationalize the denominator.y 2xHypotenuse 2 multiplied by shorter leg.y 2(4 3 )Substitute 4 3for x.y 8 3Simplify.Find the values of x and y. Give your answers in simplest radical form.5.

XY6.

XYx 9; y 93x 2 3 ; y 47. XYqi8. ##### Special Operations Mastery Event June 16. Review Money Tasks Vs. Optimization Tasks Special Operations Mastery Event 2013
Documents ##### Study Guide for Content Mastery - Student Edition - Glencoeglencoe.com/sites/california/student/science/assets/pdfs/sgcm2.pdf · Study Guide for Content Mastery Student Edition.
Documents ##### MPM1D EXAM REVIEW: MASTERY CONCEPTS - phsmath9phsmath9.wikispaces.com/file/view/The+Beast-+Exam+Review.pdf · MPM1D EXAM REVIEW: MASTERY ... 300 is 75? b) 220 is ... Estimate the
Documents ##### SCRIPTURE MASTERY - Church of Jesus Christ · 2015. 1. 14. · MASTERY SCRIPTURE MASTERY SCRIPTURE MASTERY SCRIPTURE MASTERY SCRIPTURE MASTERY SCRIPTURE MASTERY SCRIPTURE MASTERY
Documents ##### Realistic Pencil Portrait Mastery Review - The Pros and Cons
Documents ##### Study Guide for Content Mastery - Quia · PDF fileStudy Guide for Content Mastery Earth Science: Geology, the Environment, and the Universevii STUDY GUIDE FOR CONTENT MASTERY Search
Documents ##### Optical Coherence Tomography - The Quick Guide … Quick Guide to OCT Mastery - Vol... · OPTICAL COHERENCE TOMOGRAPHY The Quick Guide to OCT Mastery 50 Real Cases with Expert Analysis
Documents ##### AP English Week of Sept. 15. Mastery Review Correct Mastery Review ●Attend to corrections ●Ask questions of Mr. Conrad or of each other ●Be sure you know
Documents ##### Study Guide for Content Mastery - Student Edition · iv Chemistry: Matter and Change Study Guide for Content Mastery This Study Guide for Content Mastery for Chemistry: Matter and
Documents ##### California Standards LESSON Review for Mastery 9-1 ...iceffdams.sharpschool.net/UserFiles/Servers/Server_30085...California Standards 21.0, 23.0 Review for Mastery Solving Quadratic
Documents Documents