Transcript 4.2: Angle Relationships in Triangles

Welcome Geometry!
• Take out your homework
• Take out your whiteboard and whiteboard pens.
• Take out a piece of paper and title it:
– 4.2: Angle Relationships in Triangles
Transformation Test
• A: 28-31
• B: 25- 27.5
• C: 22-24.5
Test Corrections
• This will count as a 5 point homework
assignment due TOMORROW
Whiteboards: DO NOW
Classify each triangle by its angles and sides.
1. MNQ
2. NQP
3. MNP
Whiteboards
1. Find the measure of exterior DBA of BCD,
if mDBC = 30°, mC= 70°, and mD = 80°
2. What is the complement of an angle with
measure 17°?
4.2: Angle Relationships in Triangles
• Learning Objective
– SWBAT find the measures and apply theorems of
interior and exterior angles of triangles.
MATH JOKE OF THE DAY
• How many feet are in a yard?
• It depends on how many people are in the
yard!
Materials
Patty paper
Straightedge
Piece of paper split in half
Pencil/eraser
Directions
1. Draw and label triangle ABC on your paper
2. On patty paper, draw a line l and label a point
P on the line
3. Place line l on AB and place point P on angle
B of your triangle. Trace angle B.
Directions
1. Rotate the triangle until point P is on Angle C
and trace angle C. It should be adjacent to
Angle B.
2. Rotate the triangle again and trace angle A
adjacent to angle C.
Answer the following Question on
your Notes:
1. What do you notice about the three angles of
the triangle?
Now do it Again…
• With a different size triangle
• What do you observe about your results?
• Write an equation describing the relationship
among the measures of the interior angles in a
triangle.
• This is the Triangle Sum Theorem
• Tape your triangle and patty paper in your
notebooks.
Triangle Sum Theorem
An auxiliary line is a line that is added to a figure to aid in a
proof.
4
C
Y
2
1
X
5
An auxiliary line used in the
Triangle Sum Theorem
2
A
3
B
Talk with your group…
• What is the relationship between angles 1 and 4?
• What is the relationship between angles 3 and 5?
• Using the angle addition postulate, what do angles 1,
2 and 3 equal?
C
4
2
A
X
1
5
3
B
Proofs are Back!!!!
Example 1: Application
After an accident, the positions of cars are
measured by law enforcement to
investigate the collision.
1. Use the diagram drawn from the
information collected to find mXYZ.
2. Find mYWZ
mXYZ + mYZX + mZXY = 180°
Whiteboards
• Use the diagram
to find mMJK.
A corollary is a theorem whose proof follows directly from
another theorem. Here are two corollaries to the Triangle Sum
Theorem.
Example 2: Finding Angle Measures in
Right Triangles
One of the acute angles in a right triangle
measures 2x°. What is the measure of the other
acute angle?
Let the acute angles be A and B, with mA = 2x°.
mA + mB = 90°
2x + mB = 90
mB = (90 – 2x)°
Whiteboards
• The measure of one of the acute angles in a
right triangle is x°. What is the measure of
the other acute angle?
• Interior
• all points inside the figure
• Exterior
• all points outside the figure.
1.
What are the interior angles?
2.
What are the exterior angles?
Exterior
Interior
Exterior Angle Theorem
The measure of an exterior angle of a
triangle is equal to the sum of the
measures of its non-adjacent interior
angles
4= 1 + 2
Example 3: Applying the Exterior Angle
Theorem
Find mB.
Whiteboards
Find mACD.
Third Angle Theorem
Example 4: Applying the Third Angle
Theorem
Find mK and mJ.
Whiteboards
Find mP and mT.
Whiteboards
1. The measure of one of the acute angles in a right triangle is 56 °. What
is the measure of the other acute angle?
2. Find mABD.
124°
3. Find mN and mP.
75°; 75°
WHITEBOARDS
4. The diagram is a map showing John's house, Kay's house, and
the grocery store. What is the angle the two houses make with the
store?