1/15/20144-2: 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

04/10/23 4-2: Measuring Angles in Triangles

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.


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.


Justify the sum of the measures of A, B and C.∠ ∠ ∠

AB

C


Triangle Sum Theorem:

The sum of the measures of the angles of a triangle is _______°.


Determine the value of x.

A

C

B58 x - 8

x

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?

A.40°

B.60°

C.80°

D.120°

E.160°


A

B

C


Calculate the measures of each angle below.

7x+3 5x+9

8x-12

A

C

B

04/10/23 4-2: Measuring Angles in TrianglesA

C

B

F

ED

If ∠A≅∠D and ∠C≅∠F, what may you conclude about ∠B and ∠E?

Can you justify your conjecture?



Third Angle Congruence Theorem:

If two angles of one triangle are _____________ to 2 angles of a second triangle, then the third angles are also _____________ .


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.


Exterior Angles

A B

C

D

∠CBD is an exterior angle for ∆ABC


Exterior Angles

A B

C

E

∠ABE is an exterior angle for ∆ABC


Exterior Angles

Because there are 2 exterior angles at each vertex, every triangle has a total of 6 exterior angles.


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.


A C

B

D

Remote Interior Angles:∠ __ and ∠__ are remote interior angles for exterior ∠BCD.


A C

B

E

Remote Interior Angles:∠__ and ∠__ are remote interior

angles for exterior ∠BAE.


Exterior Angle Theorem:

The measure of an exterior angle of a triangle is the sum of the measures of


Exterior Angle Theorem: A Proof

A C

B

D

Given: ΔABC with exterior angle∠BCDProve: m∠BCD = m∠A + m∠B


Given: ΔABC with exterior angle ∠BCDProve: m∠BCD = m∠A + m∠B

AC

B

D


What is the measure of ∠A?

124°

A

52°


A corollary is a theorem that is easily proven by another theorem.

Corollary


Acute Angles Corollary

The acute angles of a right triangle are ____________.

A

C B

m∠A + m∠B =


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.


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.



If AB is perpendicular to BC, determine the measures of all 8 numbered angles.


Assignment:

pages 193-195,

# 15-19 (odds), 36, 39, 42-44 (all)

Documents

1/15/20144-2: 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