Adjacent and Vertical Angles
In this activity, you will learn about angle pairs that are formed when two lines intersect.
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In the Gizmotm, under
Conditions,
select None. Be sure that Adjacent is selected.
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Notice
AOB
and
BOC
. What do you observe about the vertex of each angle? What else do these angles share?
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Angles that share a vertex and have a common side but no common interior points are called adjacent angles. Drag A, B, and C to see other pairs of adjacent angles. On paper, make a sketch of two adjacent angles.
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Under
Conditions,
select Vertical angles.
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Vertical angles are pairs of angles that are formed by intersecting lines and are opposite each other. Name two pairs of vertical angles in the diagram shown in the Gizmo.
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Click on Click to measure angles to turn on the Gizmo protractors. (For help using the protractors, click on Gizmo Help, below the Gizmo.) Use the protractors to measure
AOB
,
BOC
,
COD
, and
AOD
. Record your data. Click on Show angle measures to check your work.
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You should have found that the pairs of vertical angles in this diagram are
AOB
and
COD
, and
BOC
and
AOD
. What do you notice about the measures of these pairs of vertical angles? Make a conjecture about vertical angles.
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Drag A, B, C, and D to see more angles formed by two intersecting lines. Do vertical angles always seem to be congruent? Explain, and provide sketches of some of the angles you formed.
Complementary and Supplementary Angles
In this activity, you will learn about angle pairs whose measures add to
90°
or
180°
.
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Under
Conditions,
select Complementary to congruent angles. Be sure that Adjacent is selected. Adjust the angles by dragging points B and E.
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Two angles are complementary if their measures add up to
90°
. Name two pairs of adjacent complementary angles in the diagram in the Gizmo. (You can also say that the angles are complements of each other.)
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Select Show angle measures. What is
mDYE
? What is the sum of
mDYE
and
mEYF
? Use this information to find
mEYF
. Then turn on a Gizmo protractor and measure
EYF
to check your answer.
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In general, if two angles are complementary and you know the measure of one of the angles, how can you find the measure of the second angle? Drag point E to form new angles and test your method. Write down one example that demonstrates your method. Does it always work?
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Complementary angles can be adjacent or nonadjacent. Select Nonadjacent in the Gizmo. Change the angles by dragging points E and F. Name two pairs of complementary angles. Turn on Show angle measures and use the protractor to confirm that the angles are complementary.
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It is given that
AWB
and
EYF
are congruent. What other pair of angles is congruent?
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Under
Conditions,
select Complementary to same angle. Name two pairs of complementary angles. (Hint: Look for overlapping right angles.)
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Turn on Show angle measures. Drag A, B, C, and D to form new angles. What two angles are always congruent?
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Use what you know about complementary angles to find
mBOC
. Then use the protractor to check your answer.
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Under
Conditions,
select Supplementary to same angle. With Adjacent selected, adjust the angles by dragging points A, B, C, and D.
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Two angles are supplementary if their measures add up to
180°
. Name two different angles that are supplementary to
BOC
. (You can also say that the angles are supplements of each other.)
-
Select Show angle measures. What is
mAOB
? What is the sum of
mAOB
and
mBOC
? Use this information to find
mBOC
. Then turn on a Gizmo protractor and measure
BOC
to check your answer.
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In general, if two angles are supplementary and you know the measure of one of the angles, how can you find the measure of the second angle? Drag any of the points to form new angles and test your method. Write down one example that demonstrates your method. Does it always work?
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Supplementary angles can be adjacent or nonadjacent. Select Nonadjacent in the Gizmo. Change the angles by dragging points A and B. Name the two pairs of supplementary angles.