Download Warm-Up Exercises 1.

Warm-Up Exercises
Classify each angle as acute, obtuse, or right.
1. 90º
ANSWER
right
2. 72º
ANSWER
acute
3. 116º
ANSWER
obtuse
4. How do you know that
~ 2?
1=
2
ANSWER
Alt. Int. s Thm.
1
Warm-Up Exercises
Target
Classify Triangles by Sides and
Angles
You will…
Classify triangles and find measures
of their angles.
Warm-Up Exercises
Vocabulary
• triangle – a polygon with 3 sides
To classify by sides:
• scalene – no congruent sides
• isosceles – at least 2 congruent sides
• equilateral – exactly 3 congruent sides
To classify by angles:
• acute – exactly 3 acute angles
• right – exactly 1 right angle
• obtuse – exactly 1 obtuse angle
• equiangular – exactly 3 congruent angles
Warm-Up Exercises
Vocabulary
• interior angles – original angles of the triangle – “inside”
• exterior angles – the angles that form linear pairs with
the interior angles of a triangle
Warm-Up Exercises
Vocabulary
Triangle Sum Theorem 4.1 –
the sum of the measures of the interior angles of any
triangle is 180°
Corollary – the acute angles of a right
triangle are complementary
Warm-Up Exercises
Vocabulary
Exterior Angle Theorem 4.2 –
the measure of an exterior angle of a triangle is equal to
the sum of the measures of the two non-adjacent
interior angles
m A+ m
B=m
1
Warm-Up2Exercises
EXAMPLE
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 =
5
2.2
=
45
6.7
( 6 – (– 1 )) 2 + ( 3 – 2 ) 2 =
50
7.1
OQ =
( x2 – x1 ) 2 + ( y2 – y1 ) 2
=
( 6 – 0 )2 + ( 3 – 0 )2
PQ =
( x2 – x1 ) 2 + ( y2 – y1 ) 2
=
Warm-Up2Exercises
EXAMPLE
Classify a triangle in a coordinate plane
STEP 2 Check for right angles.
2–0
The slope of OP is
= – 2.
– 1– 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.
Warm-Up
Exercises
GUIDED
PRACTICE
1.
A
for Examples 1 and 2
Draw an obtuse isosceles triangle and an acute
scalene triangle.
B
C
obtuse isosceles triangle
Q
R
P
acute scalene triangle
Warm-Up
Exercises
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 right.
BC =
( ( 3 ) – 0 ) 2 + ( 3 – 0 ) 2 = 18
( ( 3 ) – 3 ) 2 + ( −3 – 3 ) 2 = 36
AC =
(( 0 ) – 3 )2 + ( 0 – 3 )2 =
AB =
3–0
= 1.
3–0
3–3
The slope of BC is
= 0.
−3 – 3
3–0
The slope of AC is
= −1.
−3 – 0
The slope of AB is
ANSWER
ABC
is isosceles
18
BAC
is a right angle
ABC is a right Isosceles triangle.
Warm-Up3Exercises
EXAMPLE
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°
x = 75
Apply the Exterior Angle Theorem.
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°.
Warm-Up4Exercises
EXAMPLE
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° .
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 3 and 4
3. Find the measure of
1 in the diagram shown.
ANSWER The measure of  1 in the diagram is 65°.
Warm-Up
Exercises
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
m
A+m
B+m
C=
180°
Apply Triangle
Sum Theorem
x + 2x + 3x = 180°
6x = 180°
x = 30°
B : 2x = 2(30) = 60°
C : 3x = 3(30) = 90°
Warm-Up
Exercises
GUIDED
PRACTICE
for Examples 3 and 4
5. Find the measures of the acute
angles of the right triangle in the
diagram shown.
ANSWER
26° and 64°
6. In Example 4, what is the
measure of the obtuse angle
formed between the staircase
and a segment extending from
the horizontal leg?
ANSWER
ACD =150°.
A
2x
B
x
C
Q