Geometry: Proving Angles Are Congruent
Proving Angles Are Congruent
Two angles are congruent if they have the same measure. You already know that when two lines intersect the vertical angles formed are congruent. You have also seen that if ∠A and ∠B are each complementary to ∠C, then ∠A ~= ∠B. There are other angle relationships to explore. When you expose these angle relationships, you will establish their truth using a formal proof.
For example, you were introduced to the idea of an angle bisector. Well, it turns out that the bisector of an angle divides the angle into two angles, each of which has measure equal to one-half the measure of the original angle.
This statement looks a lot like Theorem 9.1 applied to angles rather than segments. You can use a game plan similar to the one you used to prove Theorem 9.1 to prove this theorem.
- Example 3: Prove that the bisector of an angle divides the angle into two angles, each of which has measure equal to one-half the measure of the original angle.
- Solution: Go step-by-step through the formal proof.
- 1. Give a statement of the theorem.
- Theorem 9.3: The bisector of an angle divides the angle into two angles, each of which has measure equal to one-half the measure of the original angle.
- 2. Draw a picture. You need an angle and its bisector. Figure 9.3 shows ∠ABC bisected by →BD.
Figure 9.3∠ABC is bisected by →BD.
- 3. State what is given in terms of your drawing. You are given ∠ABC which is bisected by →BD.
- 4. State what you want to prove in terms of your drawing. You want to prove that m∠ABD = 1/2 m∠ABC.
- 5. Write the proof. You must think about which definitions, postulates, and theorems you can make use of. The first one that comes to mind is the definition of an angle bisector. The postulate that will come in handy is the Angle Addition Postulate, which states that if a point D lies in the interior of ∠ABC, then m∠ABC + m∠DBC = m∠ABC. That's most of what you'll need to cook up this proof. You'll need a pinch of algebra to complete the dish.
| Statements | Reasons | |
|---|---|---|
| 1. | →BD is the angle bisector of ∠ABC | Given |
| 2. | ∠ABD ~= ∠DBC | Definition of angle bisector |
| 3. | m∠ABD = m∠DBC | Definition of |
| 4. | m∠ABC + m∠DBC = m∠ABC | Angle Addition Postulate |
| 5. | 2m∠ABD = m∠ABC | Substitution (steps 3 and 4) |
| 6. | m∠ABD = 1/2 m∠ABC | Algebra |
Excerpted from The Complete Idiot's Guide to Geometry © 2004 by Denise Szecsei, Ph.D.. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with Alpha Books, a member of Penguin Group (USA) Inc.
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