Transcript 4.2: Angle Relationships in Triangles

4.2: Angle Relationships in Triangles
Objectives:
• Find the measures of interior and exterior angles of
triangles.
• Apply theorems about the interior and exterior angles of
triangles.
Supplies:
Math Notebook
Textbook
Assignment: p. 228 #15-23(skip 17), 29-32
4.2: Angle Relationships in Triangles
•A line drawn connected to a shape to
help write a proof.
•In this case I have extended a side of
a triangle to create an exterior angle.
An angle on the inside of a shape.
An angle that is created by extending a
side of a shape.
(in this picture, and the next, it is the angle mark in
black)
•The two angles not connected to the
created exterior angle.
•Another way to say this is that they are
the angles opposite the exterior angle.
(In the picture the turquoise and purple angles.)
4.2: Angle Relationships in Triangles
∠A+∠B+∠C=180°
∠A+∠B=∠D
If ∠A=∠E and ∠B=∠F then ∠ =∠
Corollaries to Triangle Sum Theorem
H
4-2-2: The acute angles of a right triangle are complementary.
4-2-3: If a triangle is equiangular, then each angle measures 60°.
K
∠H + ∠K=90°
4.2: Angle Relationships in Triangles
After an accident, the positions of cars are measured by
law enforcement to investigate the collision. Use the
diagram drawn from the information collected to find
mXYZ.
mXYZ + mYZX + mZXY = 180°
mXYZ + 40 + 62 = 180
mXYZ + 102 = 180
mXYZ = 78°
Sum. Thm
Substitute Property
Simplify.
Subtract 102 from both sides.
4.2: Angle Relationships in Triangles
After an accident, the positions of cars are measured by
law enforcement to investigate the collision. Use the
diagram drawn from the information collected to find
mYWZ.
180-(62+40)=180-102=87
180- (40+(12+87))=180-(40+99)=180-139=74
I just solved it! Can you
rewrite my work in proof
form and provide justification
for each step?
1 Bonus point on the next quiz
if you can and do.
4.2: Angle Relationships in Triangles
The measure of one of the acute angles in a right triangle is 63.7°. What is the
measure of the other acute angle?
Statement
∠A+∠B=90°
∠A+∠63.7° =90°
∠A =26.3°
Proof
Acute angles of a right triangle are complementary
Substitution property
Subtraction property of equality
The measure of one of the acute angles in a right triangle is 48  . What is the
measure of the other acute angle?
90°-48 °=42
If I asked you to write a proof and give you space on a
quiz.
Which one of these answers would get all 6 points?
How many points would you give someone for the other
answer?
4.2: Angle Relationships in Triangles
Find mB.
Given
Justification
m∠A+m∠B= m∠BCD
Exterior Angles Theorem
15+2x+3= 5x-60
Substitution property
2x+18= 5x-60
Simplification
78=3x
Subtraction property of equality
26=x
Division property of equality
m∠B= 2x+3
Given
m∠B= 2(26)+3
Substitution
m∠B= 55°
Simplification.
4.2: Angle Relationships in Triangles
Find mACD.
Statement
Justification
m∠ABC +m∠BAC =m∠ACD
90°+(2z+1)°=(6z-9)°
2z+91=6z-9
100=4z
25=z
m∠ACD=(6z-9)
m∠ACD=6(25)-9
m∠ACD=141°
I provided the statements,
If You provide
justifications for each step,
Then you will receive
1 Bonus point on the next
quiz.
4.2: Angle Relationships in Triangles
Find mK and mJ.
Statement
Justification
m∠FKH= m∠IJG
Third angles theorem
4y²=6y²-40
Substitution property
40=2y²
Subtraction property of equality
20= y²
Division property of equality
m∠FKH=4y²
Given
m∠FKH=4(20)
Substitution
m∠FKH=80
Simplification
m∠IJG=80
Transitive property of equality
4.2: Angle Relationships in Triangles
Assignment:
p. 228: 15-23(skip 17), 29-32