Download Angles of Triangles

Angles of Triangles
3-4
EXAMPLE 1
Classify triangles by sides and by angles
Support Beams
Classify the triangular shape of the support beams in
the diagram by its sides and by measuring its 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.
EXAMPLE 2
Classify a triangle in a coordinate plane
Classify PQO by its sides.
Then determine if the triangle
is a right triangle.
SOLUTION
STEP 1 Use the distance formula to find the side lengths.
OP =
=
( x2 – x1 ) 2 + ( y2 – y1 ) 2
( (– 1 ) – 0 ) 2 + ( 2 – 0 ) 2 =
OQ =
( x2 – x1 ) 2 + ( y2 – y1 ) 2
=
( 6 – 0 )2 + ( 3 – 0 )2
=
5
2.2
45
6.7
EXAMPLE 2
Classify a triangle in a coordinate plane
PQ =
=
( x2 – x1 ) 2 + ( y2 – y1 ) 2
( 6 – (– 1 )) 2 + ( 3 – 2 ) 2 =
50
7.1
STEP 2 Check for right angles.
2–0
The slope of OP is
= – 2.
–2–0
The slope of OQ is 3 – 0 = 1 .
2
6–0
1
The product of the slopes is – 2
= – 1,
2
so OP
OQ and
ANSWER
Therefore,
POQ is a right angle.
PQO is a right scalene triangle.
for Examples 1 and 2
GUIDED PRACTICE
1.
Draw an obtuse isosceles triangle and an acute
scalene triangle.
B
Q
A
C
obtuse isosceles triangle
R
P
acute scalene triangle
GUIDED PRACTICE
2.
for Examples 1 and 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 =
=
BC =
=
( x2 – x1 ) 2 + ( y2 – y1 ) 2
(( 3 ) – 0 )2 + ( 3 – 0 )2 =
( x2
18
4.2
= 400
20
x1 ) 2 + ( y2 – y1 ) 2
( –3 – 3) 2 + ( 3 – 3 ) 2
GUIDED PRACTICE
AC =
=
for Examples 1 and 2
( x2 – x1 ) 2 + ( y2 – y1 ) 2
( (–3 – 0 ) ) 2 + ( 3 – 0 ) 2 =
18
4.2
STEP 2 Check for right angles.
The slope of AB is 3 – 0 = 1.
3–0
The slope of AC is 3 – 0 = – 1 .
–3–0
The product of the slopes is 1(– 1) = – 1 ,
so AB
AC and
ANSWER
Therefore,
BAC is a right angle.
ABC is a right Isosceles triangle.
EXAMPLE 3
ALGEBRA
Find an angle measure
Find m JKM.
SOLUTION
STEP 1 Write and solve an equation to find the value
of x.
(2x – 5)° = 70° + x° Apply the Exterior Angle Theorem.
x = 75
Solve for x.
STEP 2 Substitute 75 for x in 2x – 5 to find m JKM.
2x – 5 = 2 75 – 5 = 145
ANSWER The measure of  JKM is 145°.
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.
x° + 2x° = 90°
x = 30
Corollary to the Triangle Sum Theorem
Solve for x.
ANSWER
So, the measures of the acute angles are 30° and
2(30°) = 60° .
GUIDED PRACTICE
for Examples 3 and 4
3. Find the measure of
1 in the diagram shown.
SOLUTION
STEP 1 Write and solve an equation to find the value
of x.
(5x – 10)° =40° + 3x° Apply the Exterior Angle Theorem.
2x = 50
x= 25
Solve for x.
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°
ANSWER So measure of  1 in the diagram is 65°.
GUIDED PRACTICE
for Examples 3 and 4
4. Find the measure of each interior angle of
ABC,
where m A = x °, m B = 2x° , and m C = 3x°.
SOLUTION
A+ B+
x
C = 180°
x + 2x + 3x = 180°
6x = 180°
x = 30°
B = 2x = 2(30) = 60°
C = 3x = 3(30) = 90°
2x
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.
(x – 6)° + 2x° = 90°
Corollary to the Triangle Sum Theorem
3x = 96
x = 32
Solve for x.
Substitute 32 for x in equation x – 6 = 32 – 6 = 26°.
ANSWER So, the measure of acute angle 2(32) = 64°
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
2x
SOLUTION
x
Q
B
C
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.
GUIDED PRACTICE
for Examples 3 and 4
Use the corollary to set up and solve an equation.
x° + 2x = 90°
x = 30
Corollary to the Triangle Sum Theorem
Solve for x.
So the measures of the acute angles are
30° and 2(30°) = 60°
ACD is linear pair to
So 30° +
ACD.
ACD = 180°.
ANSWER Therefore =
ACD = 150°.