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

4.1 Triangles and Angles 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 obtuse x = 5; 8; 16; 23

Objectives 4.1 Triangles and Angles 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.

Vocabulary 4.1 Triangles and Angles acute triangle Corollary equiangular triangle Legs right triangle Adjacent obtuse triangle Exterior equilateral triangle Interior isosceles triangle Hypotenuse scalene triangle base

4.1 Triangles and Angles 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 C A B AB, BC, and AC are the sides of ABC. A, B, C are the triangle's vertices.

4.1 Triangles and Angles In a triangle, two sides sharing a common vertex are Adjacent sides.

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.

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.

4.1 Triangles and Angles Acute Triangle Three acute angles Triangle Classification By Angle Measures Acute Triangle Three acute angles

Three congruent acute angles 4.1 Triangles and Angles Triangle Classification By Angle Measures Equiangular Triangle Three congruent acute angles

4.1 Triangles and Angles Right Triangle One right angle Triangle Classification By Angle Measures Right Triangle One right angle

4.1 Triangles and Angles Obtuse Triangle One obtuse angle Triangle Classification By Angle Measures Obtuse Triangle One obtuse angle

Example 1A: Classifying Triangles by Angle Measures 4.1 Triangles and Angles Example 1A: Classifying Triangles by Angle Measures Classify BDC by its angle measures. B is an obtuse angle. B is an obtuse angle. So BDC is an obtuse triangle.

Example 1B: Classifying Triangles by Angle Measures 4.1 Triangles and Angles Example 1B: Classifying Triangles by Angle Measures Classify ABD by its angle measures. ABD and CBD form a linear pair, so they are supplementary. Therefore mABD + mCBD = 180°. By substitution, mABD + 100° = 180°. So mABD = 80°. ABD is an acute triangle by definition.

4.1 Triangles and Angles Check It Out! Example 1 Classify FHG by its angle measures. EHG is a right angle. Therefore mEHF +mFHG = 90°. By substitution, 30°+ mFHG = 90°. So mFHG = 60°. FHG is an equiangular triangle by definition.

4.1 Triangles and Angles Equilateral Triangle Three congruent sides Triangle Classification By Side Lengths Equilateral Triangle Three congruent sides

At least two congruent sides 4.1 Triangles and Angles Triangle Classification By Side Lengths Isosceles Triangle At least two congruent sides

4.1 Triangles and Angles Scalene Triangle No congruent sides Triangle Classification By Side Lengths Scalene Triangle No congruent sides

4.1 Triangles and Angles 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 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.

4.1 Triangles and Angles

4.1 Triangles and Angles 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

4.1 Triangles and Angles 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 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

Example 1A: Application 4.1 Triangles and Angles Example 1A: Application 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 mXYZ. mXYZ + mYZX + mZXY = 180° Sum. Thm Substitute 40 for mYZX and 62 for mZXY. mXYZ + 40 + 62 = 180 mXYZ + 102 = 180 Simplify. mXYZ = 78° Subtract 102 from both sides.

Example 1B: Application 4.1 Triangles and Angles Example 1B: Application 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 mYWZ. 118° Step 1 Find mWXY. mYXZ + mWXY = 180° Lin. Pair Thm. and  Add. Post. 62 + mWXY = 180 Substitute 62 for mYXZ. mWXY = 118° Subtract 62 from both sides.

Example 1B: Application Continued 4.2 Congruence and Triangles Example 1B: Application Continued 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 mYWZ. 118° Step 2 Find mYWZ. mYWX + mWXY + mXYW = 180° Sum. Thm Substitute 118 for mWXY and 12 for mXYW. mYWX + 118 + 12 = 180 mYWX + 130 = 180 Simplify. mYWX = 50° Subtract 130 from both sides.

4.1 Triangles and Angles Check It Out! Example 1 Use the diagram to find mMJK. mMJK + mJKM + mKMJ = 180° Sum. Thm Substitute 104 for mJKM and 44 for mKMJ. mMJK + 104 + 44= 180 mMJK + 148 = 180 Simplify. mMJK = 32° Subtract 148 from both sides.

4.1 Triangles and Angles A corollary is a theorem whose proof follows directly from another theorem. Here are two corollaries to the Triangle Sum Theorem.

Example 2: Finding Angle Measures in Right Triangles 4.1 Triangles and Angles Example 2: Finding Angle Measures in Right Triangles One of the acute angles in a right triangle measures 2x°. What is the measure of the other acute angle? Let the acute angles be A and B, with mA = 2x°. mA + mB = 90° Acute s of rt. are comp. 2x + mB = 90 Substitute 2x for mA. mB = (90 – 2x)° Subtract 2x from both sides.

4.1 Triangles and Angles Check It Out! Example 2a The measure of one of the acute angles in a right triangle is 63.7°. What is the measure of the other acute angle? Let the acute angles be A and B, with mA = 63.7°. mA + mB = 90° Acute s of rt. are comp. 63.7 + mB = 90 Substitute 63.7 for mA. mB = 26.3° Subtract 63.7 from both sides.

4.1 Triangles and Angles Check It Out! Example 2b The measure of one of the acute angles in a right triangle is x°. What is the measure of the other acute angle? Let the acute angles be A and B, with mA = x°. mA + mB = 90° Acute s of rt. are comp. x + mB = 90 Substitute x for mA. mB = (90 – x)° Subtract x from both sides.

4.2 Congruence and Triangles Check It Out! Example 2c The measure of one of the acute angles in a right triangle is 48 . What is the measure of the other acute angle? 2° 5 Let the acute angles be A and B, with mA = 48 . 2° 5 mA + mB = 90° Acute s of rt. are comp. 48 + mB = 90 2 5 Substitute 48 for mA. 2 5 mB = 41 3° 5 Subtract 48 from both sides. 2 5

4.2 Congruence and Triangles The interior is the set of all points inside the figure. The exterior is the set of all points outside the figure. Exterior Interior

4.1 Triangles and Angles 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. 4 is an exterior angle. Exterior Interior 3 is an interior angle.

4.1 Triangles and Angles Each exterior angle has two remote interior angles. A remote interior angle is an interior angle that is not adjacent to the exterior angle. 4 is an exterior angle. The remote interior angles of 4 are 1 and 2. Exterior Interior 3 is an interior angle.

4.1 Triangles and Angles

Example 3: Applying the Exterior Angle Theorem 4.1 Triangles and Angles Example 3: Applying the Exterior Angle Theorem Find mB. mA + mB = mBCD Ext.  Thm. Substitute 15 for mA, 2x + 3 for mB, and 5x – 60 for mBCD. 15 + 2x + 3 = 5x – 60 2x + 18 = 5x – 60 Simplify. Subtract 2x and add 60 to both sides. 78 = 3x 26 = x Divide by 3. mB = 2x + 3 = 2(26) + 3 = 55°

4.2 Congruence and Triangles Check It Out! Example 3 Find mACD. mACD = mA + mB Ext.  Thm. Substitute 6z – 9 for mACD, 2z + 1 for mA, and 90 for mB. 6z – 9 = 2z + 1 + 90 6z – 9 = 2z + 91 Simplify. Subtract 2z and add 9 to both sides. 4z = 100 z = 25 Divide by 4. mACD = 6z – 9 = 6(25) – 9 = 141°

Example 2A: Classifying Triangles by Side Lengths 4.1 Triangles and Angles Example 2A: Classifying Triangles by Side Lengths Classify EHF by its side lengths. From the figure, . So HF = 10, and EHF is isosceles.

Example 2B: Classifying Triangles by Side Lengths 4.1 Triangles and Angles Example 2B: Classifying Triangles by Side Lengths Classify EHG by its side lengths. By the Segment Addition Postulate, EG = EF + FG = 10 + 4 = 14. Since no sides are congruent, EHG is scalene.

4.1 Triangles and Angles Check It Out! Example 2 Classify ACD by its side lengths. From the figure, . So AC = 15, and ACD is isosceles.

Example 3: Using Triangle Classification 4.1 Triangles and Angles Example 3: Using Triangle Classification Find the side lengths of JKL. Step 1 Find the value of x. Given. JK = KL Def. of  segs. Substitute (4x – 10.7) for JK and (2x + 6.3) for KL. 4x – 10.7 = 2x + 6.3 Add 10.7 and subtract 2x from both sides. 2x = 17.0 x = 8.5 Divide both sides by 2.

4.1 Triangles and Angles Example 3 Continued Find the side lengths of JKL. Step 2 Substitute 8.5 into the expressions to find the side lengths. JK = 4x – 10.7 = 4(8.5) – 10.7 = 23.3 KL = 2x + 6.3 = 2(8.5) + 6.3 = 23.3 JL = 5x + 2 = 5(8.5) + 2 = 44.5

4.1 Triangles and Angles Check It Out! Example 3 Find the side lengths of equilateral FGH. Step 1 Find the value of y. Given. FG = GH = FH Def. of  segs. Substitute (3y – 4) for FG and (2y + 3) for GH. 3y – 4 = 2y + 3 Add 4 and subtract 2y from both sides. y = 7

Check It Out! Example 3 Continued 4.1 Triangles and Angles Check It Out! Example 3 Continued Find the side lengths of equilateral FGH. Step 2 Substitute 7 into the expressions to find the side lengths. FG = 3y – 4 = 3(7) – 4 = 17 GH = 2y + 3 = 2(7) + 3 = 17 FH = 5y – 18 = 5(7) – 18 = 17

4.1 Triangles and Angles Example 4: Application A steel mill produces roof supports by welding pieces of steel beams into equilateral triangles. Each side of the triangle is 18 feet long. How many triangles can be formed from 420 feet of steel beam? The amount of steel needed to make one triangle is equal to the perimeter P of the equilateral triangle. P = 3(18) P = 54 ft

Example 4: Application Continued 4.1 Triangles and Angles Example 4: Application Continued A steel mill produces roof supports by welding pieces of steel beams into equilateral triangles. Each side of the triangle is 18 feet long. How many triangles can be formed from 420 feet of steel beam? To find the number of triangles that can be made from 420 feet of steel beam, divide 420 by the amount of steel needed for one triangle. 420  54 = 7 triangles 7 9 There is not enough steel to complete an eighth triangle. So the steel mill can make 7 triangles from a 420 ft. piece of steel beam.

4.1 Triangles and Angles Check It Out! Example 4a Each measure is the side length of an equilateral triangle. Determine how many 7 in. triangles can be formed from a 100 in. piece of steel. The amount of steel needed to make one triangle is equal to the perimeter P of the equilateral triangle. P = 3(7) P = 21 in.

Check It Out! Example 4a Continued 4.1 Triangles and Angles Check It Out! Example 4a Continued Each measure is the side length of an equilateral triangle. Determine how many 7 in. triangles can be formed from a 100 in. piece of steel. To find the number of triangles that can be made from 100 inches of steel, divide 100 by the amount of steel needed for one triangle. 100  7 = 14 triangles 2 7 There is not enough steel to complete a fifteenth triangle. So the manufacturer can make 14 triangles from a 100 in. piece of steel.

4.1 Triangles and Angles Check It Out! Example 4b Each measure is the side length of an equilateral triangle. Determine how many 10 in. triangles can be formed from a 100 in. piece of steel. The amount of steel needed to make one triangle is equal to the perimeter P of the equilateral triangle. P = 3(10) P = 30 in.

Check It Out! Example 4b Continued 4.1 Triangles and Angles Check It Out! Example 4b Continued Each measure is the side length of an equilateral triangle. Determine how many 10 in. triangles can be formed from a 100 in. piece of steel. To find the number of triangles that can be made from 100 inches of steel, divide 100 by the amount of steel needed for one triangle. 100  10 = 10 triangles The manufacturer can make 10 triangles from a 100 in. piece of steel.

4.1 Triangles and Angles Lesson Quiz I 1. MNQ 2. NQP Classify each triangle by its angles and sides. 1. MNQ 2. NQP 3. MNP 4. Find the side lengths of the triangle. acute; equilateral obtuse; scalene acute; scalene 29; 29; 23

4.1 Triangles and Angles Lesson Quiz: Part II 5. The measure of one of the acute angles in a right triangle is 56 °. What is the measure of the other acute angle? 6. Find mABD. 2 3 33 ° 1 3 124°

4.1 Triangles and Angles Lesson Quiz: Part III 7. The diagram is a map showing John's house, Kay's house, and the grocery store. What is the angle the two houses make with the store? 30°