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Properties of Angles and Triangles
In this unit, you will
● identify relationships among the measures of angles formed by parallel lines cut by a transversal.
● also prove deductively, the properties of angles in a triangle and other polygons.
● explore the different ways to prove that two triangles are congruent.
● then use this information to complete geometric proofs.
Previous Knowledge
1. Sum of the angles of a triangle equal 180°– This will be proven later
Previous Knowledge
2. Perpendicular and parallel lines– Perpendicular lines intersect at a right angle, 90°
– Parallel lines never intersect and are the same distance apart from each at all points
● Where could you see perpendicular lines in real life– Corners, desks, floors and walls
● Where could you see parallel lines in real life– Tiles, window with blinds, parking lot lines
New Stuff
● Transversal– A line that crosses at least two other lines at
specific points
● How many angles are formed when a transversal intersects two parallel lines?
Properties of Angles
● Measure all the angles and note the patterns● Which angles are equal?● Which angles are supplementary?● Is this inductive reasoning?●
Different types of Angles
● Vertically Opposite Angles– Angles that are opposite to each other when two
lines cross
– Vertically opposite angles are equal
– Angles a & b are vertically opposite angles
–
● Corresponding Angles– When two lines are intersected by a transversal, the
angles in matching corners are corresponding angles
– Corresponding angles are equal
– a/e, b/f, c/g, d/h are corresponding angles
● Alternate Interior Angles– Pairs of angles on opposite sides of the transversal,
but are inside the two lines
– Alternate interior angles are equal
– c/f and d/e are alternate interior angles
● Alternate Exterior Angles– The pairs of angles on opposite sides of the
transversal, but outside the two lines
– Alternate exterior angles are equal
– a/h, b/g are alternate exterior angles
● Consecutive Interior Angles– The pairs of angles on one side of the transversal,
but inside the two lines
– Consecutive Interior Angles are supplementary (add up to 180°)
– d/f, c/e are consecutive interior angles
Note
● Consecutive Interior Angles are called interior angles on the same side of the transversal in the textbook
How to tell if two lines are parallel
Examples
Summary of Angles
● Vertically Opposite Angles are equal● Corresponding Angles are equal● Alternate Interior/Exterior Angles are equal● Consecutive Interior Angles are supplementary
● Name one pair of angles that are alternate exterior
● Name one pair of angles that are corresponding angles
● Name one pair of angles that are vertically opposite angles
● s and z, t and y are alternate exterior● s and w, u and y, t and x, v and z are
corresponding angles● s and v, t and u, w and z, x and y are vertically
opposite angles
● Find the measure of ALL the missing angles
● Are these lines parallel?
● No they are not b/c the angles shown are corresponding angles and they are not equal
● Are these lines parallel?
● Yes there are b/c the angles shown are consecutive interior angles and they are supplementary (they add up to 180 degrees)
● Are these lines parallel?
● No they are not b/c alternate exterior angles are not equal
For you to do!
● Pg. 72 #'s 5, 6
Notes on Notation● When writing angles we sometimes use 3 points,
with the middle point being the vertex of the angle● An angle symbol, , is placed in front of the 3
points● AXZ or ZXA NOT ZAX
● On occasion, we use only the point at the vertex if there will be no confusion as to what angle we are looking at
● For parallel lines, when we say line AB is parallel to line CD, we use two vertical lines to indicate they are parallel
Section 2.2: Angles formed by parallel lines
Drawing parallel lines using a compass
● Draw a straight line and a point (P) NOT on the line
● Draw a line through P intersecting the line at a point (Q)– The exact angle is not important
● Using a compass, construct an arc centered at Q and passing through both lines. Label the intersection points R and S.
● Draw another arc, centered at P, with the same radius as the arc RS. Label the intersection point (T).
● Draw a third arc, with center T, and same radius as RS that intersects the previous arc drawn. Label the intersection point W.
● Draw a line that goes through P and W.
● To show that lines PW and QS are parallel, just show that one of the following angle properties is true.
● Find the missing angles. Justify your answers.
● x = 62 (corresponding angles)● y = 118 (supplementary angles)● z = 62 (vertically opposite angles with x)
● Find the missing angles. Justify your answer.
● x = 125 (supplementary angles)● y = 55 (alternate interior angles or consecutive
interior angles are supplementary)● Z = 125 (corresponding angles)
● A = 110 (corresponding angle)● B = 110 (vertically opposite angle with a)● C = 70 (consecutive interior angles are
supplementary)● D = 70 (alternate interior angles)
● Determine the value of x
● They are alternate exterior angles and thus equal– 2x – 10 = x + 15
– x = 25
● Fix the errors in the following problem
● Here's what it should look like.
Proving conjectures using two-column proofs
● Conjecture:
When a transversal intersects a pair of parallel lines, the alternate interior angles are equal.
● Conjecture:– When a transversal intersects a pair of parallel
lines, the alternate exterior angles are equal.
Example
● Determine the missing angles
● Conjecture:– When a transversal intersects a pair of parallel
lines, the interior angles on the same side as the transversal are supplementary
● Given the situation below, would you expect Prince Phillip Drive and Elizabeth Avenue to ever cross if they went on forever?
For you to do!
● Pg.78-82 #'s: 1, 2, 3, 4, 15, 20
Section 2.3: Angles Properties in Triangles
Proving the sum of interior angles of triangles is 180°
● Draw a triangle● Draw a line which is parallel to one of the sides
of the triangle and tangent at one of the vertices
● Identify pairs of equal angles
● What is the sum of the measures of <1, <2, <3?● <1 + <2 + <3 = 180° (Straight line)● Prove <2 + <4+ <5 = 180°
● Determine the measure of <P
● (15x - 4) + (4x + 5) + 65 = 180● 19x + 66 = 180● 19x = 114● X = 6
● <P– 15(6) – 4 = 86°
Exterior Angles
● In the diagram, <MTH is an exterior angle of ΔMAT
● Determine the missing angles
● <T = 180 – 155 = 25°● <A = 180 – 25 – 40 = 115°
Note
● The measure of any exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles
● <d = <a + <b
Examples
● What is measure of <XYC?
● Find the measures of a & b
Section 2.4: Angle Properties in Polygons
● Convex polygon – Each interior angle is less than 180°
● Concave polygon– One or more interior angle(s) is greater than 180°
How is the number of sides in a polygon related to the sum of its
interior angles?
Polygon Number of sides (n)
Number of triangles
Angle Sum (S)
Triangle 3 1 1(180) = 180
Quadrilateral 4 2 2(180) = 360
Pentagon 5 3 3(180) = 540
Hexagon 6 4 4(180) = 720
Heptagon 7 5 5(180) = 900
... ... ... ...
decagon 10 8 8(180) = 1440
100-gon 100 ? ?
n-gon n n-2 (n-2) x 180
The Sum of the Interior Angles of a Polygon:
S = (n-2) x 180°● Where n = number of sides
Examples● Determine the sum of the interior angles in the
following polygons● A) B)
A) n = 8 B) n = 12
S = 180(8-2) S = 180(12-2)
S = 180(6) S = 180(10)
S = 1080° S = 1800°
● Determine the number of sides of a polygon if the sum of its interior angles is 1620°
S = 180(n-2)
1620 = 180(n-2)
9 = n-2
n = 11
The polygon has 11 sides
● Here's home plate. It has 3 right angles and two congruent angles (A & B). What are they?
● Find the sum of its interior angles
S = 180(5-2) = 540°
● Subtract from the total the right angles
540 – 3(90) = 270°
● Since A = B, half of 270 is their angle
270 = 135°
2
A = B = 135°
● Regular polygon– All sides are the same length and all interior angles
are equal
– If this isn't the case, it's an irregular polygon
Examples
● What is the measure of each interior angle of the following regular polygons?
A) B)
● A) B)
S = 180(4-2) S = 180(5-2)
S = 360° S = 540°
360 = 90° 540 = 108°
4 5
Each interior Each interior angle
angle is 90° is 108°
● Find the measures of x and y
● S = 180(10-2) (Find the total angle sum)● S = 1440°● 1440 = 144° (Find each interior angle)
10● 180 – 144 = 36° (Find the sum of x and y)● 36 = 18° = x = y (Find x and y)
2● x and y equal 36°
Summary of results
● Interior angle sum of a convex polygon– (n – 2) x 180
● Interior angle of a convex polygon– (n-2) x 180
n
Section 2.5: Exploring Congruent Triangles
Section 2.5 – 2.6: Exploring Congruent Triangles
and proving triangles are
congruent
The idea of congruence
● Two geometric figures with exactly the same size and shape
Congruent or similar?
● The two shapes need to be the same size to be congruent
● When you need to alter one shape to make it the same as the other, then those shapes are similar
Congruent?
● Why such a funny word that basically means “equal”?– They would only be “equal” if laid on top of each
other
– Comes from the Latin word congruere which means “to agree”
– So shapes “agree” when they are congruent
Congruent Triangles
● If two triangles are congruent, they will have exactly the same three sides and angles
● They may not be in the same position, but they will be there
Same sides
● If all 3 sides are the same, then triangles are congruent
● For example:
But...
● The following triangles are not congruent b/c they do not have the same sides.
Same Angles
● Doesn't always work with angles● Two triangles can have the same angles, but be
different sizes– All the angles can match, but sides are different
lengths
They could be congruent if they are the same size
● In this case, they happen to be the same size● Having the same angle is no guarantee
triangles are congruent
Definition of Congruent Triangles
● Two triangles are congruent if and only if (IFF) their corresponding parts are congruent
● CPCTC● Corresponding Parts of Congruent Triangles
are Congruent
Corresponding Parts
● Side-Side-Side● Angle-Side-Angle● Side-Angle-Side● Angle-Angle-Side
● To prove two triangles are congruent, one only needs to prove one of the above relationships is true
Side-Side-Side Congruence
● If the sides of one triangle are the same as the sides of another triangle, then the triangles are congruent
http://www.mathopenref.com/congruentsss.html
Angle-Side-Angle Congruence● If two angles and the included side of one triangle
are the same as two angles and the included side of another triangle, then the triangles are congruent
● These triangles are congruent
http://www.mathopenref.com/congruentasa.html
Side-Angle-Side Congruence● If two sides and the included angle of a triangle are
the same as two sides and the included angle of another triangle, then the triangles are congruent
● The following triangles are congruent
http://www.mathopenref.com/congruentsas.html
Angle-Angle-Side Congruence● If two angles and the non-included side are the same
as two angles and the non-included side of another triangle, then the triangles are congruent
● The following triangles are congruent
http://www.mathopenref.com/congruentaas.html
The Congruence Postulates
● Side-Side-Side (SSS)● Angle-Side-Angle (ASA)● Side-Angle-Side (SAS)● Angle-Angle-Side (AAS)
Name that Pokemon...I mean Postulate!
For you to do
● Pg. 106 #'s 1,2,3,4
2.6: Proving congruent triangles
Steps for proving triangles are congruent
● 1. Mark the givens● 2. Mark any reflexive sides, vertical angles,...● 3. Choose a method (SSS, SAS, ASA, AAS)● 4. List the parts... in the order of the method● 5. Fill in the reasons...why you marked the parts
● Pg. 113 #'s 1,2
That's all folks!
