Angles of Triangles
Section 4.1 (pg 282)
Name ____________________________
Interior Angles of a Triangle
Triangle AngleSum Theorem
The sum of the measures of the (interior) angles of a triangle is 180.
In the figure at the right, m∠ A + m∠ B + m∠ C = 180.
Example: Find m∠T.
m∠R + m∠S + m∠T = 180
25 + 35 + m∠T = 180
60 + m∠T = 180
m∠T = 120
Triangle Angle - Sum Theorem
Substitution
Simplify
Subtraction PoE
Find the measure of each numbered angle.
1.
2.
3.
4.
5.
6.
7.
8.
Section 4.1
page 1
Find the value of x and the measure of each angle.
9.
10.
11.
12.
Section 4.1
page 2
1.
Find m∠B
and m∠C.
Find the value of x and the measure of each indicated angle.
2. The angles in a triangle are represented by (4x - 6)º, (2x + 1)º and (x + 3)º. Is
this a right triangle?
A proof of the Triangle Angle-Sum Theorem requires the use of an auxiliary line. An auxiliary line is an extra line or
segment drawn in a figure to help analyze geometric relationships.
Here is one proof of the Triangle Angle-Sum Theorem.
Section 4.1
page 2
Statement
Given: ΔABC with AD∥BC
Prove:
m∠1 + m∠2 + m∠3 = 180°
Reason
1. Δ ABC with AD∥BC
Given
2. ∠1 ≅ ∠4, ∠2 ≅ ∠5
Alternate Interior Angles Theorem
3. m∠1 = m∠4, m∠2 = m∠5
Definition of Congruence
4. m∠4 + m∠ CAD = 180°
Linear Pair Postulate
5. m∠3 + m∠5 = m∠ CAD
Angle Addition Postulate
6. m∠4 + m∠3 + m∠5 = 180°
Substitution PoE
7. m∠1 + m∠3 + m∠2 = 180°
Substitution PoE
Below is a different proof of the Triangle Sum Theorem.
** Notice that angles 2 and 3 are in different places than in the previous proof!
Section 4.1
page 2
Proof Using An Auxiliary Parallel Line
Statements
Reasons
1. Δ ABC
1. Given
2. Draw auxiliary line through B parallel to
.
2. Through a point not on a line, only one line may be
drawn parallel to a given line.
3. ∠ DBE is a straight angle.
3. A straight line forms a straight angle.
4. m∠ DBE = 180º
4. A straight angle has a measure of 180º.
5. m∠1 + m∠2 + m∠3 = m∠ DBE
5. Angle Addition Postulate
6.
6. alternate interior angles converse.
7. m∠1 = m∠A; m∠3 = m∠C
7. Congruent angles are angles of equal measure.
8. m∠A + m∠2 + m∠C = 180º
8. Substitution
Slightly different version:
Proof Using An Auxiliary Parallel Line with an Extension
Statements
1. ΔABC
2. Extend
Reasons
1. Given
through C to E, and draw an auxiliary
line through C parallel to
.
2. Through a point not on a line, only one line may be
drawn parallel to a given line.
3. ∠ACE is a straight angle.
3. A straight line forms a straight angle.
4. m∠ ACE = 180º
4. A straight angle has a measure of 180º.
5. m∠1 + m∠2 + m∠3 = m∠ ACE
5. Angle Addition Postulate (whole quantity)
6.
6. alternate interior angles Theorem.
7.
7. corresponding angles Theorem
8. m∠ B = m∠2; m∠3 = m∠ A
8. Congruent angles are angles of equal measure.
9. m∠1 + m∠ B + m∠ A = 180º
9. Substitution
Section 4.1
page 2
Exterior Angles of a Triangle
An Exterior Angle is the angle formed by one side of a polygon and the
extension of the adjacent side.
In all polygons, there are two sets of exterior angles, one that goes around
clockwise and the other goes around counterclockwise.
Notice that the interior angle and its adjacent exterior angle form a linear pair and add up to 180°.
m∠1 + m∠2 = 180°
For each exterior angle of a triangle, the remote interior angles are the interior angles that are not adjacent to that
exterior angle. In the diagram below, ∠B and ∠A are the remote interior angles for exterior ∠DCB.
Exterior Angle
Theorem
The measure of an exterior angle of a triangle is equal to the sum of the
measures of the two remote interior angles. m∠1 = m∠ A + m∠ B
Example 1: Find m∠1.
m∠1 = m∠ R + m∠ S
m∠1 = 60 + 80
m∠1 = 140
Example 2: Find x.
Exterior Angle Theorem
Substitution
Simplify
m∠ PQS = m∠ R + m∠ S
78 = 55 + x
23 = x
Exterior Angle Theorem
Substitution
Subtract 55 from each side
Find the measures of each numbered angle.
1.
4.
7.
Section 4.1
3.
2.
5.
6.
8.
page 2
Find each measure.
5. Find m∠ ABC
6. Find m∠ F
Find the measures of each indicated angle.
Determine the SUM of the three exterior angles in the previous 3 problems.
Conclusion: ____________________________________________________________________________________
Find the value of x and the measure of each angle.
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
Applications
1. Sam, Kendra, and Tony are standing in such a way that if lines were drawn to
connect the friends they would form a triangle.
If Sam is looking at Kendra he needs to turn his head 40° to look at Tony. If Tony
is looking at Sam he needs to turn his head 50° to look at Kendra. How many
degrees would Kendra have to turn her head to look at Tony if she is looking at
Sam?
2.
A lookout tower sits on a network of struts and posts. Leslie measured two angles on the tower.
What is the measure of angle 1?
Section 4.1
page 3
3. Chloe bought a drafting table and set it up so that she can draw comfortably whi;e
standing up. Chloe measures the two angles created by the legs and the tabletop in
case she has to dismantle the table at some point.
a. Which of the four numbered angles can Chloe determine by knowing the two
angles formed with the tabletop?
What are their measures?
b. What conclusion can Chloe make about the unknown angles before she measures them to find their
exact measurements?
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ __ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _
Using algebra to find angle measurements involving triangles.
In ∆ QRS, mQ = x°, mR = (8x-40)°, and mS = 2x°.
Find all the angle measures in ∆ QRS.
What type of triangle is QRS?
In triangle ABC, m∠ A is twice m∠ B,
and m∠ C is 8 more than m∠ B.
What is the measure of each angle?
Write and solve an equation.
Let x = m∠ B.
m∠ A + m∠ B + m∠ C = 180
2x + x + (x + 8) = 180
4x + 8 = 180
4x = 172
x = 43
So... m∠ A = 2(43) or 86,
m∠ B = 43, and
m∠ C = 43 + 8 or 51
How could you easily check your answers?
Set up and solve an equation to answer each problem.
1. In ∆ DEF, m∠ E is three times m∠ D, and m∠ F is 9
less than m∠ E. Find the measure of each angle.
3. In ∆ JKL, m∠ K is four times m∠ J, and m∠ L is five
times m∠ J. Find the measure of each angle.
Section 4.1
2. In ∆ RST, m∠ T is 5 more than m∠ R, and m∠ S is 10
less than m∠ T. Find the measure of each angle.
4. In ∆ XYZ, m∠ Z is 2 more than twice m∠ X, and m∠ Y is
7 less than twice m∠ X. Find the measure of each
angle.
page 3
5. In ∆ GHI, m∠ H is 20 more than m∠ G, and m∠ G is
8 more than m∠ I. Find the measure of each angle.
6. In ∆ MNO, m∠ M is equal to m∠ N, and m∠ O is 5 more
than three times m∠ N. Find the measure of each
angle.
7. In ∆ STU, m∠ U is half m∠ T, and m∠ S is 30 more than m∠ T. Find the measure of each angle.
8. In ∆ PQR, m∠ P is equal to m∠ Q, and m∠ R is 24 less than m∠ P. Find the measure of each angle.
Find the angles of a triangle whose second angle exceeds the first angle by 15° and the third angle is 66° more than the
second angle.
Find the value of x and the measure of each angle.
Section 4.1
page 4
3.
Find m∠DBC.
4.
Find x.
Section 4.1
page 3
The proof of the Exterior Angle Theorem will utilize linear pairs and the sum of the
interior angles of a triangle.
Statements
1.
Reasons
1. Given
2. ∠2 and ∠4 form a linear pair
2. A linear pair is 2 adjacent ∠s whose noncommon sides form opposite rays.
3. ∠2 supp ∠4
3. If 2∠s form a linear pair, they are
supplementary.
4. m∠2 + m∠4 = 180
4. Supplementary ∠s are 2 ∠s the sum of whose
measures is 180.
5. m∠1 + m∠2 + m∠3 = 180
5. The measures of the angles of a triangle add
to 180º.
6. m∠2 + m∠4 = m∠1 + m∠2 + m∠3
6. Substitution
7. m∠2 = m∠2
7. Reflexive Property (or quantity is = itself)
8. m∠4 = m∠1 + m∠3
8. Subtraction of Equalities
Section 4.1
page 3