Download Proof writing: angle measures at a vertex

Name
September 13, 2016
Math 2 notes and homework
textbook section 6.05 page 1
Proof writing: angle measures at a vertex
Vocabulary
Angles that have equal measures are called congruent angles.
Angle addition is the idea that when angles combine to form a larger angle, the angle measures
can be added to get the measure of the larger angle.
Two angles are complementary if their measures add to 90°.
Two angles are supplementary if their measures add to 180°.
A theorem is a geometry fact that is known to be true from having proved it.
Vertical Angle Theorem
Vertical Angle Theorem: When two lines intersect, each pair of vertical
angles is congruent. (Specifically, in the diagram, 1 = 3 and 2 = 4.)
Proof of the Vertical Angle Theorem:
1 + 2 = 180°
by Angle Addition.
2 + 3 = 180°
by Angle Addition.
1 + 2 = 2 + 3 because they both equal 180° (“transitive property”).
1 = 3
by subtracting 2 from both sides.
Four more steps just like the above ones can be used to show 2 = 4.
Other ways to write this proof: Proofs can be written in varied ways, and often different steps can be
chosen that prove the same theorem. The textbook on pp. 457-458 shows another proof of this same
theorem. The important thing in any proof is that there’s a reason justifying each step.
Name
September 13, 2016
Math 2 notes and homework
textbook section 6.05 page 2
Homework
Directions:
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Write clearly enough that your work could be shown on the projector during class tomorrow.
When you are writing proofs, there needs to be a reason for each step. You can give these
reasons in your own words, but the reason has to be based on facts that are already known
(such as piece of given information or the use of an algebra step).
Three of the six problems are from the textbook. When you’re doing a textbook problem, copy
the picture onto your paper. If you do this, you won’t have to bring your textbook tomorrow.
1–2. Do problems 1 and 2 from page 459.
3. In the diagram at the right, prove that
2 = 360° – 1.
For problems 4 and 5: Use the diagram at the right and
the following given facts:
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Lines BE and DF are perpendicular at point A.
AC is an angle bisector of BAD.
4. What is the angle measure of CAE? Prove your
answer.
5. Find an angle that has a measure of 225°. Prove your
answer.
6. Do problem 6 on page 461. (The word collinear means “on the same straight line.”)
Optional challenge: Maybe you can find more than one thing to prove.