Lesson Vertical angles

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Vertical angles

In Figure 1 two intersecting straight lines are shown. They form four angles alpha

and

.
Pair of angles alpha

and

are vertical. Pair of angles gamma

and

are vertical as well.

Definition.
Two angles are called vertical if the sides of one of them are continuations of sides of the other.

Theorem (Vertical angles theorem)
Vertical angles are congruent.

Proof
Referring to Figure 1, we are going to prove that alpha

and

.
Note that
alpha

= 180° (because angles alpha

and

make in sum the straight angle), and
beta

= 180° (because angles gamma

and

make in sum the straight angle).
Hence, alpha

.

Similarly,
alpha

= 180° (because alpha

and

make in sum the straight angle) and
alpha

= 180° (because alpha

and

make in sum the straight angle).
Therefore, gamma

.
The proof is completed.

Figure 1. Vertical angles

Example
If in Figure 1 one of vertical angles alpha

= 37°,
find three other angles beta

and

Solution
beta

= 37° as the vertical angle to alpha

;

= 180° -37° = 143° as the complementary angle to alpha

;

= 143° as the vertical angle to gamma

My other lessons in this site on Angles Basics, Supplementary and Complementary angles, Vertical angles, Parallel lines are
    - Angles basics
    - Parallel lines
    - HOW TO solve problems on supplementary, complementary or vertical angles - Examples
    - HOW TO solve problems on parallel lines - Examples
    - OVERIEW of lessons on Angles Basics, Supplementary and Complementary angles, Vertical angles, Parallel lines

To navigate over the lessons on Properties of Triangles use this file/link Properties of Trianles.

To navigate over all topics/lessons of the Online Geometry Textbook use this file/link GEOMETRY - YOUR ONLINE TEXTBOOK.

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