November 7, 2018 5.1 Angles of Triangles

Geometry 5.1 Angles of Triangles

Essential Question How are the angle measures of a triangle related? November 7, 2018 5.1 Angles of Triangles

Goals – Day 1 Classify triangles by their sides Classify triangles by their angles Identify parts of triangles. Find angle measures in triangles. November 7, 2018 5.1 Angles of Triangles

Triangle Symbol Use the picture  for triangle. November 7, 2018 5.1 Angles of Triangles

This is ABC, which can also be named BCA, CAB, BAC, CBA, or ACB. Triangle A triangle is a figure formed by three segments joining three noncollinear points. B A C This is ABC, which can also be named BCA, CAB, BAC, CBA, or ACB. November 7, 2018 5.1 Angles of Triangles

Classifying Triangles by Sides Equilateral  Isosceles  Scalene  November 7, 2018 5.1 Angles of Triangles

Equilateral Triangle Three congruent sides. November 7, 2018 5.1 Angles of Triangles

Isosceles Triangle At least two congruent sides. November 7, 2018 5.1 Angles of Triangles

Scalene Triangle No congruent sides. November 7, 2018 5.1 Angles of Triangles

Classifying Triangles by Angles Right  Equiangular  Acute  Obtuse  November 7, 2018 5.1 Angles of Triangles

Right Triangle One Right Angle November 7, 2018 5.1 Angles of Triangles

Equiangular Triangle Three Congruent Angles November 7, 2018 5.1 Angles of Triangles

Acute Triangle Three acute angles November 7, 2018 5.1 Angles of Triangles

Obtuse Triangle One Obtuse Angle November 7, 2018 5.1 Angles of Triangles

And to add to the confusion… An equilateral triangle is also equiangular. An equiangular triangle is also acute. An equilateral can be considered an isosceles triangle. An equilateral triangle is also acute. November 7, 2018 5.1 Angles of Triangles

Adjacent and Opposite Sides of a Triangle Two sides that share a common vertex are adjacent sides. The third side is the opposite side from that vertex. A In RAT, RA and RT are adjacent sides. AT is the opposite side from ∠𝑅. R T November 7, 2018 5.1 Angles of Triangles

Isosceles Triangles (In this case, we consider an isosceles triangle with only two congruent sides.) The congruent sides are the LEGS. The third side is the BASE. Leg Leg Base November 7, 2018 5.1 Angles of Triangles

Right Triangle The LEGS form the right angle. The third side (opposite the right angle) is the Hypotenuse. Hypotenuse Leg Leg November 7, 2018 5.1 Angles of Triangles

Hypotenuse From the Greek “stretched against”. Always longer than either leg. November 7, 2018 5.1 Angles of Triangles

What have you learned so far? In the figure, 𝑀𝑁 ⊥ 𝑄𝑃 and 𝑀𝑃 ≅ 𝑀𝑄 . Complete the following sentence. P Q N M 1. Name the legs of the isosceles triangle PMQ. Segments PM and QM. November 7, 2018 5.1 Angles of Triangles

What have you learned so far? In the figure, 𝑀𝑁 ⊥ 𝑄𝑃 and 𝑀𝑃 ≅ 𝑀𝑄 . Complete the following sentence. P Q N M 2. Name the base of isosceles triangle  PMQ. Segment PQ. November 7, 2018 5.1 Angles of Triangles

What have you learned so far? In the figure, 𝑀𝑁 ⊥ 𝑄𝑃 and 𝑀𝑃 ≅ 𝑀𝑄 . Complete the following sentence. P Q N M 3. Name the hypotenuse of right triangle PNM. Segment PM. November 7, 2018 5.1 Angles of Triangles

What have you learned so far? In the figure, 𝑀𝑁 ⊥ 𝑄𝑃 and 𝑀𝑃 ≅ 𝑀𝑄 . Complete the following sentence. P Q N M 4. Name the legs of right triangle  PNM. Segments NP and NM. November 7, 2018 5.1 Angles of Triangles

What have you learned so far? In the figure, 𝑀𝑁 ⊥ 𝑄𝑃 and 𝑀𝑃 ≅ 𝑀𝑄 . Complete the following sentence. P Q N M 5. Name the acute angles of right triangle  QNM. Q and NMQ November 7, 2018 5.1 Angles of Triangles

Example 1 Classify these triangles by its angles and by its sides. a. 125° Obtuse , Isosceles  Equiangular, Equilateral  Isosceles , Acute  Right , Scalene  November 7, 2018 5.1 Angles of Triangles

5.1 Triangle Sum Theorem The sum of the measures of the interior angles of a triangle is 180°. A B C mA + mB + mC = 180° November 7, 2018 5.1 Angles of Triangles

Example 2 Find the measure of 1. Solution: m1 + 70 + 32 = 180 70° 32° 1 November 7, 2018 5.1 Angles of Triangles

Example 3 In MAD: mM = (2x)° mA = (3x)° mD = (4x) Find the measure of each angle, and classify. Solution: 2x + 3x + 4x = 180 9x = 180 x = 20 = 2(20) = 40° = 3(20) = 60° = 4(20) = 80° This triangle is acute. November 7, 2018 5.1 Angles of Triangles

Example 4 In RST: mR=(5x + 10) mS=(2x + 15) mT=(3x + 35) Find the measure of the three angles and then classify the triangle by angles. November 7, 2018 5.1 Angles of Triangles

ACUTE Example 4 Solution (5x + 10) + (2x + 15) + (3x + 35) = 180 mR=(5x + 10) = 5(12) + 10 = 70 mS=(2x + 15) = 2(12) + 15 = 39 mT=(3x + 35) = 3(12) + 35 = 71 November 7, 2018 5.1 Angles of Triangles

Corollary to Theorem 5.1 The acute angles of a right triangle are complementary. m1 + m2 + 90 = 180 m1 + m2 = 90 QED 1 2 November 7, 2018 5.1 Angles of Triangles

Example 5 Find X x = 70° Since this is a right triangle, the acute angles are complementary, and 90 – 20 = 70. 20° x° November 7, 2018 5.1 Angles of Triangles

Interior and Exterior Angles Start with a triangle… November 7, 2018 5.1 Angles of Triangles

1, 2, 3 are INTERIOR ANGLES. They are INSIDE the triangle. Extend the sides…. 2 1 3 1, 2, 3 are INTERIOR ANGLES. They are INSIDE the triangle. November 7, 2018 5.1 Angles of Triangles

4, 6, 8, 9, 10, and 12 are EXTERIOR ANGLES. 3 10 6 12 4, 6, 8, 9, 10, and 12 are EXTERIOR ANGLES. They are OUTSIDE the triangle. They are ADJACENT to the interior angles. November 7, 2018 5.1 Angles of Triangles

5, 7, and 11 are NOT EXTERIOR ANGLES. 2 1 3 5 11 5, 7, and 11 are NOT EXTERIOR ANGLES. They are simply vertical angles to the interior angles. November 7, 2018 5.1 Angles of Triangles

Exterior angles are always supplementary to the interior angles. It is common (and less confusing) to draw only one exterior angle at a vertex. Exterior angles are always supplementary to the interior angles. 6 3 1 2 5 4 Interior Angles: 1, 2, 3 Exterior Angles: 4, 5, 6 November 7, 2018 5.1 Angles of Triangles

5.2 Exterior Angle Theorem The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles. 1 2 3 m1 = m2 + m3 November 7, 2018 5.1 Angles of Triangles

Note: Sometimes (usually) the two nonadjacent interior angles are referred to as REMOTE INTERIOR ANGLES. The theorem then reads: An exterior angle of a triangle is equal to the sum of the two remote interior angles. November 7, 2018 5.1 Angles of Triangles

5.2 Exterior Angle Thm Proof (Informal) m2 + m3 + m4 = 180 ( angle sum) m4 + m1 = 180 (linear pair postulate) m2 + m3 + m4 = m4 + m1 (substitution) m2 + m3 = m1 (subtraction) 1 2 3 4 November 7, 2018 5.1 Angles of Triangles

Naming Remote Interior Angles 1 2 3 4 6 5 8 7 9 For exterior 1, the remote interior angles are_____________. 𝒎𝟔 + 𝒎𝟖=𝒎1 November 7, 2018 5.1 Angles of Triangles

Naming Remote Interior Angles 1 2 3 4 6 5 8 7 9 For exterior 4, the remote interior angles are_____________. 𝒎𝟐 + 𝒎𝟖=𝒎𝟒 November 7, 2018 5.1 Angles of Triangles

Naming Remote Interior Angles 1 2 3 4 6 5 8 7 9 For exterior 5, the remote interior angles are_____________. 𝒎𝟐 + 𝒎𝟖=𝒎𝟓 November 7, 2018 5.1 Angles of Triangles

Naming Remote Interior Angles 1 2 3 4 6 5 8 7 9 For exterior 9, the remote interior angles are_____________. 𝒎𝟔 + 𝒎𝟐=𝒎𝟗 November 7, 2018 5.1 Angles of Triangles

Naming Remote Interior Angles 1 2 3 4 6 5 8 7 9 For remote interior angles 6 & 8, the exterior angle is _____________. 𝒎𝟔 + 𝒎𝟖=𝒎∠𝟏=𝒎∠𝟑 November 7, 2018 5.1 Angles of Triangles

Naming Remote Interior Angles 1 2 3 4 6 5 8 7 9 For remote interior angles 2 & 6, the exterior angle is _____________. 𝒎𝟔 + 𝒎𝟐=𝒎∠𝟕=𝒎∠𝟗 November 7, 2018 5.1 Angles of Triangles

Naming Remote Interior Angles 1 2 3 4 6 5 8 7 9 For remote interior angles 2 & 8, the exterior angle is _____________. 𝒎𝟐 + 𝒎𝟖=𝒎∠𝟒=𝒎∠𝟓 November 7, 2018 5.1 Angles of Triangles

Example 6 Find m1. By Theorem 5.2: m1 + 45 = 110 45° By Theorem 5.2: m1 + 45 = 110 m1 = 110 – 45 = 65° 1 110° November 7, 2018 5.1 Angles of Triangles

Example 7 (x + 15) + 45 = 3x – 10 x + 60 = 3x – 10 70 = 2x x = 35 45° (x + 15)° (3x – 10)° Solve for x. November 7, 2018 5.1 Angles of Triangles

A Final Challenge Problem… Find the measure of each numbered angle. 40° 30° 60° 20° 1 2 3 4 5 6 7 50° 60° 60° 90° 60° 100° 60° November 7, 2018 5.1 Angles of Triangles

Problems for You Use the exterior angle theorem! Write down the equation for each problem and solve. November 7, 2018 5.1 Angles of Triangles

Your Turn. 1. Find m1 Solution: m1 = 32 + 125 m1 = 157 32 1 125 November 7, 2018 5.1 Angles of Triangles

2. Find m2 Solution: m2 + 45 = 165 m2 = 120 45 2 165 November 7, 2018 5.1 Angles of Triangles

3. Solve for x. Solution: 2x + 30 + 60 = 110 2x + 90 = 110 2x = 20 110° (2x + 30)° 60 November 7, 2018 5.1 Angles of Triangles

4. Solve for x. Solution: 12x – 4 = (6x + 8) + 5x 12x – 4 = 11x + 8 November 7, 2018 5.1 Angles of Triangles

5. Solve for x. Solution: (3x + 2) + (5x – 10) = 7x + 3 November 7, 2018 5.1 Angles of Triangles

Summary The sum of the interior angles of a triangle is 180 degrees. The acute angles of a right triangle are complementary. An exterior angle is equal to the sum of the two remote interior angles. November 7, 2018 5.1 Angles of Triangles

Assignment November 7, 2018 5.1 Angles of Triangles