4-1 Triangles and Angles Holt Geometry Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz

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  • 4-1 Triangles and Angles Holt Geometry Warm Up Warm Up Lesson Presentation Lesson Presentation Lesson Quiz Lesson Quiz
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  • Warm Up Classify each angle as acute, obtuse, or right. 1. 2. 3. 4. If the perimeter is 47, find x and the lengths of the three sides. right acute x = 5; 8; 16; 23 obtuse 4.1 Triangles and Angles
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  • Classify triangles by their angle measures and side lengths. Use triangle classification to find angle measures and side lengths. Find the measures of interior and exterior angles of triangles. Apply theorems about the interior and exterior angles of triangles. Objectives 4.1 Triangles and Angles
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  • acute triangle Corollary equiangular triangle Legs right triangleAdjacent obtuse triangleExterior equilateral triangleInterior isosceles triangleHypotenuse scalene trianglebase Vocabulary 4.1 Triangles and Angles
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  • Recall that a triangle ( ) is a polygon with three sides. Triangles can be classified in two ways: by their angle measures or by their side lengths. 4.1 Triangles and Angles
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  • B A C AB, BC, and AC are the sides of ABC. A, B, C are the triangle's vertices. 4.1 Triangles and Angles
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  • In a triangle, two sides sharing a common vertex are Adjacent sides.
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  • 4.1 Triangles and Angles In a right triangle, the two sides making the right angle are called Legs. The side opposite of the right angle is called the Hypothenuse.
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  • 4.1 Triangles and Angles In an isosceles triangle, the two sides that are congruent are called the Legs. The third side is the called the base.
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  • Acute Triangle Three acute angles Triangle Classification By Angle Measures 4.1 Triangles and Angles
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  • Equiangular Triangle Three congruent acute angles Triangle Classification By Angle Measures 4.1 Triangles and Angles
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  • Right Triangle One right angle Triangle Classification By Angle Measures 4.1 Triangles and Angles
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  • Obtuse Triangle One obtuse angle Triangle Classification By Angle Measures 4.1 Triangles and Angles
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  • Classify BDC by its angle measures. Example 1A: Classifying Triangles by Angle Measures B is an obtuse angle. B is an obtuse angle. So BDC is an obtuse triangle. 4.1 Triangles and Angles
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  • Classify ABD by its angle measures. Example 1B: Classifying Triangles by Angle Measures ABD and CBD form a linear pair, so they are supplementary. Therefore mABD + mCBD = 180. By substitution, mABD + 100 = 180. So mABD = 80. ABD is an acute triangle by definition. 4.1 Triangles and Angles
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  • Classify FHG by its angle measures. Check It Out! Example 1 EHG is a right angle. Therefore mEHF +mFHG = 90. By substitution, 30+ mFHG = 90. So mFHG = 60. FHG is an equiangular triangle by definition. 4.1 Triangles and Angles
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  • Equilateral Triangle Three congruent sides Triangle Classification By Side Lengths 4.1 Triangles and Angles
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  • Isosceles Triangle At least two congruent sides Triangle Classification By Side Lengths 4.1 Triangles and Angles
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  • Scalene Triangle No congruent sides Triangle Classification By Side Lengths 4.1 Triangles and Angles
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  • Remember! When you look at a figure, you cannot assume segments are congruent based on appearance. They must be marked as congruent. 4.1 Triangles and Angles
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  • When the sides of the triangle are extended, the original angles are the interior angles. The angles that are adjacent (next to) the interior angles are the exterior angles.
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  • 4.1 Triangles and Angles
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  • An auxiliary line is a line that is added to a figure to aid in a proof. An auxiliary line used in the Triangle Sum Theorem 4.1 Triangles and Angles
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  • A corollary is a theorem whose proof follows directly from another theorem. Here are two corollaries to the Triangle Sum Theorem. 4.1 Triangles and Angles
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  • An auxiliary line is a line that is added to a figure to aid in a proof. An auxiliary line used in the Triangle Sum Theorem 4.1 Triangles and Angles
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  • After an accident, the positions of cars are measured by law enforcement to investigate the collision. Use the diagram drawn from the information collected to find mXYZ. Example 1A: Application mXYZ + mYZX + mZXY = 180 Sum. Thm mXYZ + 40 + 62 = 180 Substitute 40 for mYZX and 62 for mZXY. mXYZ + 102 = 180 Simplify. mXYZ = 78 Subtract 102 from both sides. 4.1 Triangles and Angles
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  • After an accident, the positions of cars are measured by law enforcement to investigate the collision. Use the diagram drawn from the information collected to find mYWZ. Example 1B: Application mYXZ + mWXY = 180 Lin. Pair Thm. and Add. Post. 62 + mWXY = 180 Substitute 62 for mYXZ. mWXY = 118 Subtract 62 from both sides. Step 1 Find m WXY. 118 4.1 Triangles and Angles
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  • After an accident, the positions of cars are measured by law enforcement to investigate the collision. Use the diagram drawn from the information collected to find mYWZ. Example 1B: Application Continued Step 2 Find mYWZ. 118 mYWX + mWXY + mXYW = 180 Sum. Thm mYWX + 118 + 12 = 180 Substitute 118 for mWXY and 12 for mXYW. mYWX + 130 = 180 Simplify. mYWX = 50 Subtract 130 from both sides. 4.2 Congruence and Triangles
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  • Use the diagram to find mMJK. Check It Out! Example 1 mMJK + mJKM + mKMJ = 180 Sum. Thm mMJK + 104 + 44= 180 Substitute 104 for mJKM and 44 for mKMJ. mMJK + 148 = 180 Simplify. mMJK = 32 Subtract 148 from both sides. 4.1 Triangles and Angles
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  • A corollary is a theorem whose proof follows directly from another theorem. Here are two corollaries to the Triangle Sum Theorem. 4.1 Triangles and Angles
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  • One of the acute angles in a right triangle measures 2x. What is the measure of the other acute angle? Example 2: Finding Angle Measures in Right Triangles mA + mB = 90 2x + mB = 90 Substitute 2x for mA. mB = (90 2x) Subtract 2x from both sides. Let the acute angles be A and B, with m A = 2x. Acute s of rt. are comp. 4.1 Triangles and Angles
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  • The measure of one of the acute angles in a right triangle is 63.7. What is the measure of the other acute angle? Check It Out! Example 2a mA + mB = 90 63.7 + mB = 90 Substitute 63.7 for mA. mB = 26.3 Subtract 63.7 from both sides. Let the acute angles be A and B, with m A = 63.7. Acute s of rt. are comp. 4.1 Triangles and Angles
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  • The measure of one of the acute angles in a right triangle is x. What is the measure of the other acute angle? Check It Out! Example 2b mA + mB = 90 x + mB = 90 Substitute x for mA. mB = (90 x) Subtract x from both sides. Let the acute angles be A and B, with m A = x. Acute s of rt. are comp. 4.1 Triangles and Angles
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  • The measure of one of the acute angles in a right triangle is 48. What is the measure of the other acute angle? Check It Out! Example 2c mA + mB = 90 Acute s of rt. are comp. 2 5 Let the acute angles be A and B, with mA = 48. 2 5 Subtract 48 from both sides. 2525 Substitute 48 for mA. 2525 48 + mB = 90 2525 mB = 41 3 5 4.2 Congruence and Triangles
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  • The interior is the set of all points inside the figure. The exterior is the set of all points outside the figure. Interior Exterior 4.2 Congruence and Triangles
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  • An interior angle is formed by two sides of a triangle. An exterior angle is formed by one side of the triangle and extension of an adjacent side. Interior Exterior 4 is an exterior angle. 3 is an interior angle. 4.1 Triangles and Angles
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  • Each exterior angle has two remote interior angles. A remote interior angle is an interior angle that is not adjacent to the exterior angle. Interior Exterior 3 is an interior angle. 4 is an exterior angle. The remote interior angles of 4 are 1 and 2. 4.1 Triangles and Angles
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  • Find mB. Example 3: Applying the Exterior Angle Theorem mA + mB = mBCD Ext. Thm. 15 + 2x + 3 = 5x 60 Substitute 15 for mA, 2x + 3 for mB, and 5x 60 for mBCD. 2x + 18 = 5x 60 Simplify. 78 = 3x Subtract 2x and add 60 to both sides. 26 = x Divide by 3. mB = 2x + 3 = 2(26) + 3 = 55 4.1 Triangles and Angles
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  • Find mACD. Check It Out! Example 3 mACD = mA + mB Ext. Thm. 6z 9 = 2z + 1 + 90 Substitute 6z 9 for mACD, 2z + 1 for mA, and 90 for mB. 6z 9 = 2z + 91 Simplify. 4z = 100 Subtract 2z and add 9 to both sides. z = 25 Divide by 4. mACD = 6z 9 = 6(25) 9 = 141 4.2 Congruence and Triangles
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  • Classify EHF by its side lengths. Example 2A: Classifying Triangles by Side Lengths From the figure,. So HF = 10, and EHF is isosceles. 4.1 Triangles and Angles
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  • Classify EHG by its side lengths. Example 2B: Cla

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