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Angles of Angles of Triangles Triangles 3-4 3-4
Angles of TrianglesAngles of Triangles3-43-4
EXAMPLE 1 Classify triangles by sides and by angles
SOLUTION
The triangle has a pair of congruent sides, so it is isosceles. By measuring, the angles are 55° , 55° , and 70° . It is an acute isosceles triangle.
Support Beams
Classify the triangular shape of the support beams in the diagram by its sides and by measuring its angles.
EXAMPLE 2 Classify a triangle in a coordinate plane
SOLUTION
STEP 1 Use the distance formula to find the side lengths.
Classify PQO by its sides. Then determine if the triangle is a right triangle.
OP = y2 – y1( )2x2 – x1( )2 +
= 2 – 0( )2(– 1 ) 0( )2 +– = 5 2.2
OQ = y2 – y1( )2x2 – x1( )2 +2= – 0( )6 0( )2 +– 3 = 45 6.7
EXAMPLE 2 Classify a triangle in a coordinate plane
PQ = y2 – y1( )2x2 – x1( )2 +
3 – 2( )26( )2 +–= (– 1 ) = 50 7.1
STEP 2 Check for right angles.The slope of OP is 2 – 0
– 2 – 0 = – 2.
The slope of OQ is 3 – 0 6 – 0
= 21 .
1The product of the slopes is – 22 = – 1,
so OP OQ and POQ is a right angle.
Therefore, PQO is a right scalene triangle.ANSWER
GUIDED PRACTICE for Examples 1 and 2
1. Draw an obtuse isosceles triangle and an acute scalene triangle.
obtuse isosceles triangle
B
A C
acute scalene triangleP
Q
R
GUIDED PRACTICE for Examples 1 and 2
2. Triangle ABC has the vertices A(0, 0), B(3, 3), and C(–3, 3). Classify it by its sides. Then determine if it is a right triangle.
SOLUTION
STEP 1 Use the distance formula to find the side lengths.
AB = y2 – y1( )2x2 – x1( )2 +
= 3 – 0( )2( 3 ) 0( )2 +–
BC = y2 – y1( )2x2 x1( )2 +2= – 3( )–3
3( )2 +– 3
= 18 4.2
= 400 20
GUIDED PRACTICE for Examples 1 and 2
AC = y2 – y1( )2x2 – x1( )2 +
= 3 – 0( )2 0 )(–3( )2 +– = 18 4.2
STEP 2 Check for right angles.
The slope of AB is 3 – 0 3 – 0
= 1.
The product of the slopes is 1(– 1) = – 1,
so AB AC and BAC is a right angle.
The slope of AC is 3 – 0 – 3 – 0 = .– 1
Therefore, ABC is a right Isosceles triangle.ANSWER
EXAMPLE 3 Find an angle measure
SOLUTION
STEP 1 Write and solve an equation to find the value of x.
Apply the Exterior Angle Theorem.(2x – 5)° =70° + x°Solve for x.x = 75
STEP 2Substitute 75 for x in 2x – 5 to find m∠JKM.2x – 5 = 2 75 – 5 = 145
ALGEBRA Find m∠JKM.
The measure of ∠JKM is 145°.ANSWER
EXAMPLE 4 Find angle measures from a verbal description
ARCHITECTURE
The tiled staircase shown forms a right triangle. The measure of one acute angle in the triangle is twice the measure of the other. Find the measure of each acute angle.
SOLUTION
First, sketch a diagram of the situation. Let the measure of the smaller acute angle be x° . Then the measure of the larger acute angle is 2x° . The Corollary to the Triangle Sum Theorem states that the acute angles of a right triangle are complementary.
EXAMPLE 4 Find angle measures from a verbal description
Use the corollary to set up and solve an equation.
Corollary to the Triangle Sum Theoremx° + 2x° = 90°Solve for x.x = 30
So, the measures of the acute angles are 30° and 2(30°) = 60° .
ANSWER
GUIDED PRACTICE for Examples 3 and 4
SOLUTION
STEP 1 Write and solve an equation to find the value of x.
Apply the Exterior Angle Theorem. (5x – 10)° = 40° + 3x°Solve for x.2x =50
Find the measure of 1 in the diagram shown.3.
x=25
GUIDED PRACTICE for Examples 3 and 4
STEP 2 Substitute 25 for x in 5x – 10 to find 1.5x – 10 = 5 25– 10 = 115
1 + (5x – 10)° = 180
1 + 115° = 180°1 = 65°
So measure of ∠1 in the diagram is 65°.ANSWER
GUIDED PRACTICE for Examples 3 and 4
SOLUTION
A + B + C = 180°
x + 2x + 3x = 180°
6x = 180°
x = 30°
B = 2x = 2(30) = 60°
C = 3x = 3(30) = 90°
x
2x 3x
4. Find the measure of each interior angle of ABC, where m A = x , m B = 2x° , and m C = 3x°.°
GUIDED PRACTICE for Examples 3 and 4
5. Find the measures of the acute angles of the right triangle in the diagram shown.
SOLUTION
Use the corollary to set up & solve an equation. Corollary to the Triangle Sum Theorem(x – 6)° + 2x° = 90°
3x = 96
Solve for x.x = 32
Substitute 32 for x in equation x – 6 = 32 – 6 = 26°.
So, the measure of acute angle 2(32) = 64°ANSWER
GUIDED PRACTICE for Examples 3 and 4
6. In Example 4, what is the measure of the obtuse angle formed between the staircase and a segment extending from the horizontal leg?
A
B C Q
2x
xSOLUTIONFirst, sketch a diagram of the situation. Let the measure of the smaller acute angle be x° . Then the measure of the larger acute angle is 2x° . The Corollary to the Triangle Sum Theorem states that the acute angles of a right triangle are complementary.
GUIDED PRACTICE for Examples 3 and 4
Use the corollary to set up and solve an equation. Corollary to the Triangle Sum Theoremx° + 2x = 90°
Solve for x.x = 30
So the measures of the acute angles are 30° and 2(30°) = 60°
ACD is linear pair to ACD.
So 30° + ACD = 180°.Therefore = ACD = 150°.ANSWER
