Download Investigating Angle Theorems

Investigating Angle Theorems
Answer Key
Vocabulary: complementary angles, linear pair, supplementary angles, vertical angles
Prior Knowledge Questions (Do these BEFORE using the Gizmo.)
[Note: The purpose of these questions is to activate prior knowledge and get students thinking.
Students who do not already know the answers will benefit from the class discussion.]
1. Tony has a collection of 200 sports cards. He counts and finds that 40 of them are football
cards. What does this tell you about the rest of his collection?
The rest of his sports cards (160 cards) are not football cards.
2. Suppose Tony has only football and baseball cards. Now what can you say about the rest of
his collection? He has 160 baseball cards.
Gizmo Warm-up
In the Investigating Angle Theorems Gizmo™, you can manipulate a
dynamic figure to explore the properties of different angles.
1. In the Gizmo, select Vertical angles from the Conditions menu. You
should see two intersecting lines like the ones shown to the right.
A. Name the two pairs of angles that do not share a side. (They are nonadjacent.)
AOB and DOC
BOC and AOD
Both pairs are vertical angles.
B. Drag the points to resize the angles. What appears to always be true about the
measures of the vertical angles? The measures are always equal.
Turn on Show angle measures and continue to resize to check if this is always true.
2. Select Form a linear pair to view a linear pair of angles (adjacent angles whose noncommon sides form a straight line).
A. Name the linear pair by naming the adjacent angles. AXB and BXC
B. Adjust the angles by dragging point B. What seems to always be true about the
measures of a linear pair of angles? The sum of their measures is always 180°.
Turn on Show angle measures. Drag point B to check if this is always the case.
Get the Gizmo ready:
Activity A:
Complements and
supplements
 Under Conditions, select Complementary to
congruent angles.
 Be sure Adjacent is selected.
1. Both pairs of angles shown (AXB and BXC, and DYE and EYF) are complementary.
A. Drag points B and E to view a variety of complementary angles. What is true about
the measures of two complementary angles? The sum of their measures is 90°.
B. What must be true about AXB and DYE? They must be congruent.
Why? Both pairs add to 90°, so mAXB + mBXC = mDYE + mEYF. Because
BXC and EYF are congruent, AXB and DYE must also be congruent.
Turn on Show angle measures and drag point B to verify for a variety of angles.
C. Select Nonadjacent and drag the points. Which two angle pairs are complementary?
CXD and AWB
GZH and EYF
D. What must be true about CXD and GZH? They must be congruent.
Turn on Show angle measures. Experiment to see if this is always true.
E. What is true of any pair of angles that are complementary to congruent angles?
They are always congruent to each other.
2. Select Complementary to same angle and drag points A, B, C, and D.
A. What are the two pairs of complementary angles in this figure?
AOC and BOC
DOB and BOC
B. What must be true about AOC and DOB? They must be congruent.
Why? They are both complementary to the same angle.
Turn on Show angle measures and drag the points to verify this.
C. Select Nonadjacent and run a similar test. What is true about angles that are
complementary to the same angle? They are always congruent to each other.
(Activity A continued on next page)
Activity A (continued from previous page)
3. Select Supplementary to congruent angles. Both angle pairs shown (AXB and BXC,
and DYE and EYF) are supplementary and form linear pairs.
A. Drag points B and E to view a variety of supplementary angles. What can you say
about the measures of two supplementary angles? Their sum is 180°.
B. What must be true about AXB and DYE? They must be congruent.
Why? Both pairs add to 180°, so mAXB + mBXC = mDYE + mEYF. Because
BXC and EYF are congruent, AXB and DYE must also be congruent.
C. Select Nonadjacent and run a similar test. What is true about angles that are
supplementary to congruent angles? They are always congruent to each other.
4. Select Supplementary to same angle. Drag the points to view a variety of figures.
A. Name two pairs of supplementary angles that contain BOC.
AOB and BOC
COD and BOC
B. What must be true about AOB and COD? They must be congruent.
Why? They are both supplementary to the same angle.
Turn on Show angle measures and create a variety of figures to verify this.
C. Select Nonadjacent and run a similar test. What is true about angles that are
supplementary to the same angle? They are always congruent to each other.
5. Select Vertical angles and turn on Show angle measures. Drag point A until AOB is a
right angle.
A. What is true about the four angles formed? All four angles formed are right angles
and are congruent. Experiment to see if this is always true.
B. Explain why this is always the case. The right angle (AOB) is part of two linear
pairs, so each adjacent angle must also measure 90°. Also, AOB and COD are
vertical angles, so they are congruent. Thus, all four angles are right angles.
Activity B:
Using angle
concepts
Get the Gizmo ready:
 Select Supplementary and congruent under
Conditions.
1. Drag the points to see several pairs of angles that are supplementary and congruent.
A. What is true about the measures of angles that are supplementary and congruent?
Their measures are the same.
Turn on Show angle measures to check. Then, select Nonadjacent to check that
this also applies to nonadjacent angles.
B. In the space to the right,
use algebra to show why
both angles must
measure 90°.
mAXB + mCXB = 180°
2mAXB = 180°
mAXB = 90°
mCXB = 90°
2. Solve each problem. Show all of your work. Then, if possible, check in the Gizmo.
A. Suppose AXB and BXC are
complementary and congruent.
What are their measures?
C. Find the measures
AOC and DOB.
62°
mAXB + mBXC = 90°
2mAXB = 90°
mAXB = 45°
mBXC = 45°
mAOC + mBOC = 90°
mAOC + 62° = 90°
mAOC = 28°
mDOB = 28°
B. Suppose AXB and BXC form a
linear pair. If AXB is a right
angle, what is mBXC?
D. Find the values of
x and y.
50 + 4x + 10 = 180
mAXB + mBXC = 180°
4x + 60 = 180
90° + mBXC = 180°
4x = 120
mBXC = 90°
x = 30
2y = 50
y = 25
2y°
50°
(4x + 10)°