# 4-2: Measuring Angles in Triangles

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4-2: Measuring Angles in Triangles
Expectations: G1.2.1: Prove that the sum of the angle measures of a triangle is 180 and that the measure of an exterior angle of a triangle is the sum of the measures of the 2 remote interior angles. G1.2.2: Solve problems involving angle measure and side length of triangles. 3/25/2017 4-2: Measuring Angles in Triangles

4-2: Measuring Angles in Triangles
Linear Triples ∠A, ∠B and ∠C form a linear triple iff ∠A is adjacent to ∠B, ∠B is adjacent to ∠C, ∠A and ∠C have no points in common (other than the vertex) and the sum of their measures is 180. ∠A ∠B ∠C ∠A, ∠B and ∠C form a linear triple. 3/25/2017 4-2: Measuring Angles in Triangles

Justify the sum of the measures of ∠A, ∠B and ∠C.
3/25/2017 4-2: Measuring Angles in Triangles

4-2: Measuring Angles in Triangles
Triangle Sum Theorem: The sum of the measures of the angles of a triangle is _______°. 3/25/2017 4-2: Measuring Angles in Triangles

Determine the value of x.
C x A x - 8 B 58 3/25/2017 4-2: Measuring Angles in Triangles

4-2: Measuring Angles in Triangles
In triangle ABC below, the measure of ∠A is 20° and the measure of ∠B is 3 times larger than the measure of ∠C. What is the measure of angle B? 40° 60° 80° 120° 160° B C A 3/25/2017 4-2: Measuring Angles in Triangles

Calculate the measures of each angle below.
8x-12 7x+3 5x+9 A B 3/25/2017 4-2: Measuring Angles in Triangles

4-2: Measuring Angles in Triangles
If ∠A≅∠D and ∠C≅∠F, what may you conclude about ∠B and ∠E? Can you justify your conjecture? C F A B D E 3/25/2017 4-2: Measuring Angles in Triangles

4-2: Measuring Angles in Triangles
3/25/2017 4-2: Measuring Angles in Triangles

4-2: Measuring Angles in Triangles
Third Angle Congruence Theorem: If two angles of one triangle are _____________ to 2 angles of a second triangle, then the third angles are also _____________ . 3/25/2017 4-2: Measuring Angles in Triangles

4-2: Measuring Angles in Triangles
Exterior Angles An angle is an exterior angle of a triangle iff it is the union of one side of the triangle and the extension of the other side that shares the same vertex as the first side. 3/25/2017 4-2: Measuring Angles in Triangles

4-2: Measuring Angles in Triangles
Exterior Angles C ∠CBD is an exterior angle for ∆ABC A B D 3/25/2017 4-2: Measuring Angles in Triangles

4-2: Measuring Angles in Triangles
Exterior Angles C ∠ABE is an exterior angle for ∆ABC A B E 3/25/2017 4-2: Measuring Angles in Triangles

4-2: Measuring Angles in Triangles
Exterior Angles Because there are 2 exterior angles at each vertex, every triangle has a total of 6 exterior angles. 3/25/2017 4-2: Measuring Angles in Triangles

4-2: Measuring Angles in Triangles
Remote Interior Angles: A remote interior angle of a triangle is one of the 2 interior angles not adjacent to an exterior angle. Remote interior angles change depending on which interior angle is being discussed. 3/25/2017 4-2: Measuring Angles in Triangles

4-2: Measuring Angles in Triangles
Remote Interior Angles: ∠ __ and ∠__ are remote interior angles for exterior ∠BCD. B A C D 3/25/2017 4-2: Measuring Angles in Triangles

4-2: Measuring Angles in Triangles
Remote Interior Angles: ∠__ and ∠__ are remote interior angles for exterior ∠BAE. B E A C 3/25/2017 4-2: Measuring Angles in Triangles

4-2: Measuring Angles in Triangles
Exterior Angle Theorem: The measure of an exterior angle of a triangle is the sum of the measures of 3/25/2017 4-2: Measuring Angles in Triangles

Exterior Angle Theorem: A Proof
Given: ΔABC with exterior angle ∠BCD Prove: m∠BCD = m∠A + m∠B B A C D 3/25/2017 4-2: Measuring Angles in Triangles

Given: ΔABC with exterior angle ∠BCD Prove: m∠BCD = m∠A + m∠B
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4-2: Measuring Angles in Triangles
What is the measure of ∠A? 124° A 52° First hour start here 4/19/05 3/25/2017 4-2: Measuring Angles in Triangles

4-2: Measuring Angles in Triangles
Corollary A corollary is a theorem that is easily proven by another theorem. 3/25/2017 4-2: Measuring Angles in Triangles

Acute Angles Corollary
The acute angles of a right triangle are ____________. A C B m∠A + m∠B = 3/25/2017 4-2: Measuring Angles in Triangles

Obtuse Angle Corollary
Right Angle Corollary There can be at most ___ right angle in a triangle. Obtuse Angle Corollary There can be at most ___ obtuse angle in a triangle. 3/25/2017 4-2: Measuring Angles in Triangles

4-2: Measuring Angles in Triangles
In triangle ABC, m∠A is 16 more than m∠B and m∠C is 29 more than m∠B. Determine the measure of each angle. 3/25/2017 4-2: Measuring Angles in Triangles

4-2: Measuring Angles in Triangles
3/25/2017 4-2: Measuring Angles in Triangles

4-2: Measuring Angles in Triangles
If AB is perpendicular to BC, determine the measures of all 8 numbered angles. 3/25/2017 4-2: Measuring Angles in Triangles

4-2: Measuring Angles in Triangles
Assignment: pages , # (odds), 36, 39, (all) 3/25/2017 4-2: Measuring Angles in Triangles