Vertical angles are the opposite angles formed by two intersecting lines. Vertical angles are congruent because the angles have the same measure.
Adjacent angles are pairs of angles that share a vertex and one side but do not overlap.
Complementary angles are two angles whose measures have a sum of 90°.
Supplementary angles are two angles whose measures have a sum of 180°. We discovered in the Explore Activity that adjacent angles formed by two intersecting lines are supplementary.
Example 1 :
Look at the diagram given below and answer the question.
A. Name a pair of vertical angles :
Vertical angles are opposite angles formed by intersecting lines.
∠AFB and ∠DFE are vertical angles.
B. Name a pair of adjacent angles :
Adjacent angles share a vertex and a side but do not overlap.
∠AFB and ∠BFD are adjacent angles.
C. Name a pair of supplementary angles :
Adjacent angles formed by intersecting lines are supplementary.
∠AFB and ∠BFD are supplementary angles.
D. Name two pairs of supplementary angles that include <DFE :
Any angle that forms a line with ∠DFE is a supplementary angle to ∠DFE.
∠DFE and ∠EFA are supplementary angles, as are ∠DFE and ∠DFB.
E. Find the measure of angle <AFB :
Use the fact that ∠AFB and ∠BFD in the diagram are supplementary angles to find m∠AFB.
∠AFB + ∠BFD = 180°
From the given diagram, we have ∠AFD = x and also
m∠BFD = 50° + 90° = 140°
x + 140° = 180°
Subtract 140° from both sides.
(x + 140°) - 140° = 180° - 140°
x = 40°
The measure of m∠AFB is 40°.
Example 2 :
Find the value of 'x' in the diagram given below.
From the picture above, it is very clear (x+7)° and 100° are vertical angles and also they are congruent.
So, we have
x + 7 = 100
Subtract 7 from both sides
(x + 7) - 7 = 100 - 7
x = 93
So, the value of 'x' is 93.
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