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Glencoe Geometry: Online Textbook Help13 chapters | 152 lessons

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Lesson Transcript

Instructor:
*Elizabeth Foster*

Elizabeth has been involved with tutoring since high school and has a B.A. in Classics.

In this lesson, you'll learn to prove angle relationships using properties of congruent, complementary, and supplementary angles. Learn the concepts and apply them on practice problems.

Do proofs occasionally feel a little bitâ€¦complicated? In this lesson, we'll break it down and start with a few of the basic relationships. We'll start from the beginning with the properties of congruent angles. Understanding these rules can help you build a foundation for using more complicated theorems and properties. Then we'll kick it up a notch by adding complementary and supplementary angles to the mix.

**Congruent angles** are angles with the same measure. So for example, if you have two 62 degree angles, both are congruent.

Congruent angles have various properties that can help you do proofs with them:

- The
**reflexive property**states that an angle is congruent to itself. This one is confusing if you overthink it, but there's no secret meaning; there really is a rule in geometry literally saying that something is equal to itself. - The
**symmetric property**states that if angle*A*equals angle*B*, then angle*B*equals angle*A*. It's called symmetric because the quantities on both sides of the equals sign are the same, so the equation is symmetrical. You can flip*A*and*B*from side to side, and it doesn't matter. - The
**transitive property**states that if angle*A*equals angle*B*, and if angle*B*equals angle*C*, then angle*A*equals angle*C*.

Next up, some theorems about supplementary and complementary angles. Complementary angles add up to 90 degrees, or a right angle. Supplementary angles add up to 180 degrees, which is a straight line.

The **complement theorem** states that angles complementary to the same angle are congruent to each other. In this drawing (see video), angle A and angle *B* are both complementary to 64 degrees. So both angle *A* and angle *B* must be equal to 26 degrees. From this, we can say that angle *A* and angle *B* are equal to each other. This works even if we don't know what the values of any of the angles are. Even if we don't know what *x* is, we know that both *A* and *B* are equal to 90 - *x*, so they must be equal to each other.

The **supplement theorem** states that angles supplementary to the same angle are congruent to each other. It's basically the same thing as the complement theorem, but with supplementary angles instead of complementary. Just like the complement theorem, the supplement theorem also works if you don't know exactly what the angle measures are.

Now, let's put all those things together and use them in a practice proof.

**Given that D = B and E = G, prove that A = E**

We'll start with the supplement theorem. You can see that angle *E* is supplementary to angle *D*, and angle *A* is supplementary to angle *B*. That means that angle *E* is equal to angle *A*.

The question asks us to prove that *A* = *E*, so we can just flip that around with the symmetric property of congruence. Ta-dah!

In this lesson, you learned about a bunch of different properties of angles and how you can apply them to prove relationships between angles.

- The
**reflexive property of congruence**states that an angle is equal to itself. - The
**symmetric property of congruence**states that if*A*=*B*,*B*=*A*. - The
**transitive property of congruence**states that if*A*=*B*and*B*=*C*,*A*=*C*. - The
**complement theorem**states that if angle*A*and angle*B*are both complementary to the same angle, then*A*and*B*are congruent. - The
**supplement theorem**states that if angle*A*and angle*B*are both supplementary to the same angle, then*A*and*B*are congruent.

You can use these more basic properties to start working on more complicated proofs and understanding how they work.

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Glencoe Geometry: Online Textbook Help13 chapters | 152 lessons

- What Is Inductive Reasoning? - Examples & Definition 3:52
- Critical Thinking and Logic in Mathematics 4:27
- Propositions, Truth Values and Truth Tables 9:49
- Logical Math Connectors: Conjunctions and Disjunctions 3:39
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- Biconditional Statement in Geometry: Definition & Examples 5:59
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- Law of Syllogism in Geometry: Definition & Examples 6:38
- Inductive & Deductive Reasoning in Geometry: Definition & Uses 4:59
- Properties and Postulates of Geometric Figures 4:53
- Direct Proofs: Definition and Applications 7:11
- Line Segment Bisection & Midpoint Theorem: Geometric Construction 4:39
- Algebraic Laws and Geometric Postulates 5:37
- Geometric Proofs: Definition and Format 8:35
- Algebraic Proofs: Format & Examples 3:40
- Segment Addition Postulate: Definition & Examples 3:21
- Reflexive Property of Equality: Definition & Examples 3:43
- Symmetric Property of Equality: Definition & Examples 3:26
- Transitive Property of Equality: Definition & Example 3:39
- Angle Addition Postulate: Definition & Examples 5:15
- Proving Angle Relationships 3:38
- Go to Glencoe Geometry Chapter 2: Reasoning and Proof

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