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2.5: Proofs about Angle Pairs and Segments

Difficulty Level: At Grade Created by: CK-12

Learning Objectives

  • Use theorems about pairs of angles, right angles and midpoints.

Review Queue

Fill in the 2-column proof.

1. Given: \begin{align*}\overline{VX}\end{align*} is the angle bisector of \begin{align*}\angle WVY\end{align*}.

\begin{align*}\overline{VY}\end{align*} is the angle bisector of \begin{align*}\angle XVZ\end{align*}.

Prove: \begin{align*}\angle WVX \cong \angle YVZ\end{align*}

Statement Reason
1. Given
2. Definition of an Angle Bisector
3. \begin{align*}m\angle WVX = m\angle YVZ\end{align*}
4. \begin{align*}\angle WVX \cong \angle YVZ\end{align*}

Know What? The game of pool relies heavily on angles. The angle at which you hit the cue ball with your cue determines if you hit the yellow ball and if you can hit it into the pocket.

The best path to get the yellow ball into the corner pocket is to use the path in the picture to the right. You measure and need to hit the cue ball so that it hits the side of the table at a \begin{align*}50^\circ\end{align*} angle (this would be \begin{align*}m\angle 1\end{align*}). Find \begin{align*}m\angle 2\end{align*} and how it relates to \begin{align*}\angle 1\end{align*}.

If you would like to play with the angles of pool, click the link for an interactive game. http://www.coolmath-games.com/0-poolgeometry/index.html

Naming Angles

As we learned in Chapter 1, angles can be addressed by numbers and three letters, where the letter in the middle is the vertex. We can shorten this label to only the middle letter if there is only one angle with that vertex.

All of the angles in this parallelogram can be labeled by one letter, the vertex, instead of all three.

\begin{align*}& \angle MLP \ \text{is} \ \angle L && \angle LMO \ \text{is} \ \angle M\\ & \angle MOP \ \text{is} \ \angle O && \angle OPL \ \text{is} \ \angle P\end{align*}

Right Angle Theorem: If two angles are right angles, then the angles are congruent.

Proof of the Right Angle Theorem

Given: \begin{align*}\angle A\end{align*} and \begin{align*}\angle B\end{align*} are right angles

Prove: \begin{align*}\angle A \cong \angle B\end{align*}

Statement Reason
1. \begin{align*}\angle A\end{align*} and \begin{align*}\angle B\end{align*} are right angles Given
2. \begin{align*}m\angle A = 90^\circ\end{align*} and \begin{align*}m\angle B = 90^\circ\end{align*} Definition of right angles
3. \begin{align*}m\angle A = m\angle B\end{align*} Transitive \begin{align*}PoE\end{align*}
4. \begin{align*}\angle A \cong \angle B\end{align*} \begin{align*}\cong\end{align*} angles have = measures

Anytime right angles are mentioned in a proof, you will need to use this theorem to say the angles are congruent.

Same Angle Supplements Theorem: If two angles are supplementary to the same angle then the angles are congruent.

\begin{align*}m\angle A + m\angle B & = 180^\circ \ \text{and}\\ m\angle C + m\angle B & = 180^\circ\\ \text{then} \ m\angle A & = m\angle C\end{align*}

Proof of the Same Angles Supplements Theorem

Given: \begin{align*}\angle A\end{align*} and \begin{align*}\angle B\end{align*} are supplementary angles. \begin{align*}\angle B\end{align*} and \begin{align*}\angle C\end{align*} are supplementary angles.

Prove: \begin{align*}\angle A \cong \angle C\end{align*}

Statement Reason

1. \begin{align*}\angle A\end{align*} and \begin{align*}\angle B\end{align*} are supplementary

\begin{align*}\angle B\end{align*} and \begin{align*}\angle C\end{align*} are supplementary

Given

2. \begin{align*}m\angle A + m\angle B =180^\circ\end{align*}

\begin{align*}m\angle B + m\angle C = 180^\circ\end{align*}

Definition of supplementary angles
3. \begin{align*}m\angle A + m\angle B = m\angle B + m\angle C\end{align*} Substitution \begin{align*}PoE\end{align*}
4. \begin{align*}m\angle A = m\angle C\end{align*} Subtraction \begin{align*}PoE\end{align*}
5. \begin{align*}\angle A \cong \angle C\end{align*} \begin{align*}\cong\end{align*} angles have = measures

Example 1: \begin{align*}\angle 1 \cong \angle 4\end{align*} and \begin{align*}\angle C\end{align*} and \begin{align*}\angle F\end{align*} are right angles.

Which angles are congruent and why?

Solution: By the Right Angle Theorem, \begin{align*}\angle C \cong \angle F\end{align*}. Also, \begin{align*}\angle 2 \cong \angle 3\end{align*} by the Same Angles Supplements Theorem because \begin{align*}\angle 1 \cong \angle 4\end{align*} and they are linear pairs with these congruent angles.

Same Angle Complements Theorem: If two angles are complementary to the same angle then the angles are congruent.

\begin{align*}m\angle A + m\angle B & = 90^\circ \ \text{and}\\ m\angle C + m\angle B & = 90^\circ \\ \text{then} \ m\angle A & = m\angle C.\end{align*}

The proof of the Same Angles Complements Theorem is in the Review Questions. Use the proof of the Same Angles Supplements Theorem to help you.

Vertical Angles Theorem Recall the Vertical Angles Theorem from Chapter 1. We will do a proof here.

Given: Lines \begin{align*}k\end{align*} and \begin{align*}m\end{align*} intersect.

Prove: \begin{align*}\angle 1 \cong \angle 3\end{align*}

Statement Reason
1. Lines \begin{align*}k\end{align*} and \begin{align*}m\end{align*} intersect Given

2. \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 2\end{align*} are a linear pair

\begin{align*}\angle 2\end{align*} and \begin{align*}\angle 3\end{align*} are a linear pair

Definition of a Linear Pair

3. \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 2\end{align*} are supplementary

\begin{align*}\angle 2\end{align*} and \begin{align*}\angle 3\end{align*} are supplementary

Linear Pair Postulate

4. \begin{align*}m\angle 1 + m\angle 2 = 180^\circ\end{align*}

\begin{align*}m\angle 2 + m\angle 3 = 180^\circ\end{align*}

Definition of Supplementary Angles
5. \begin{align*}m\angle 1 + m\angle 2 = m\angle 2 + m\angle 3\end{align*} Substitution \begin{align*}PoE\end{align*}
6. \begin{align*}m\angle 1 = m\angle 3\end{align*} Subtraction \begin{align*}PoE\end{align*}
7. \begin{align*}\angle 1 \cong \angle 3\end{align*} \begin{align*}\cong\end{align*} angles have = measures

You can also do a proof for \begin{align*}\angle 2 \cong \angle 4\end{align*}, which would be exactly the same.

Example 2: In the picture \begin{align*}\angle 2 \cong \angle 3\end{align*} and \begin{align*}k \bot p\end{align*}.

Each pair below is congruent. State why.

a) \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 5\end{align*}

b) \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 4\end{align*}

c) \begin{align*}\angle 2\end{align*} and \begin{align*}\angle 6\end{align*}

d) \begin{align*}\angle 6\end{align*} and \begin{align*}\angle 7\end{align*}

Solution:

a) and

b) Same Angles Complements Theorem

c) Vertical Angles Theorem

d) Vertical Angles Theorem followed by the Transitive Property

Example 3: Write a two-column proof.

Given: \begin{align*}\angle 1 \cong \angle 2\end{align*} and \begin{align*}\angle 3 \cong \angle 4\end{align*}

Prove: \begin{align*}\angle 1 \cong \angle 4\end{align*}

Solution:

Statement Reason
1. \begin{align*}\angle 1 \cong \angle 2\end{align*} and \begin{align*}\angle 3 \cong \angle 4\end{align*} Given
2. \begin{align*}\angle 2 \cong \angle 3\end{align*} Vertical Angles Theorem
3. \begin{align*}\angle 1 \cong \angle 4\end{align*} Transitive \begin{align*}PoC\end{align*}

Know What? Revisited If \begin{align*}m\angle 1 = 50^\circ\end{align*}, then \begin{align*}m\angle 2 = 50^\circ\end{align*}.

Draw a perpendicular line at the point of reflection. The laws of reflection state that the angle of incidence is equal to the angle of reflection (see picture). This is an example of the Same Angles Complements Theorem.

Review Questions

Fill in the blanks in the proofs below.

  1. Given: \begin{align*}\overline{AC} \bot \overline{BD}\end{align*} and \begin{align*}\angle 1 \cong \angle 4\end{align*} Prove: \begin{align*}\angle 2 \cong \angle 3\end{align*}
Statement Reason
1. \begin{align*}\overline{AC} \bot \overline{BD}, \angle 1 \cong \angle 4\end{align*}
2. \begin{align*}m\angle 1 = m\angle 4 \end{align*}
3. \begin{align*}\bot\end{align*} lines create right angles

4. \begin{align*}m\angle ACB = 90^\circ\end{align*}

\begin{align*}m\angle ACD = 90^\circ \end{align*}

5. \begin{align*}m\angle 1 + m\angle 2 = m\angle ACB\end{align*}

\begin{align*}m\angle 3 + m\angle 4 = m\angle ACD \end{align*}

6. Substitution
7. \begin{align*}m\angle 1 + m\angle 2 = m\angle 3 + m\angle 4 \end{align*}
8. Substitution
9. Subtraction \begin{align*}PoE\end{align*}
10. \begin{align*}\angle 2 \cong \angle 3 \end{align*}
  1. Given: \begin{align*}\angle MLN \cong \angle OLP\end{align*} Prove: \begin{align*}\angle MLO \cong \angle NLP\end{align*}
Statement Reason
1.
2. \begin{align*}\cong\end{align*} angles have = measures
3. Angle Addition Postulate
4. Substitution
5. \begin{align*}m\angle MLO = m\angle NLP \end{align*}
6. \begin{align*}\cong\end{align*} angles have = measures
  1. Given: \begin{align*}\overline{AE} \bot \overline{EC}\end{align*} and \begin{align*}\overline{BE} \bot \overline{ED}\end{align*} Prove: \begin{align*}\angle 1 \cong \angle 3\end{align*}
Statement Reason
1.
2. \begin{align*}\bot\end{align*} lines create right angles

3. \begin{align*}m\angle BED = 90^\circ\end{align*}

\begin{align*}m\angle AEC = 90^\circ \end{align*}

4. Angle Addition Postulate
5. Substitution
6. \begin{align*}m\angle 2 + m\angle 3 = m\angle 1 + m\angle 3\end{align*}
7. Subtraction \begin{align*}PoE\end{align*}
8. \begin{align*}\cong\end{align*} angles have = measures
  1. Given: \begin{align*}\angle L\end{align*} is supplementary to \begin{align*}\angle M\end{align*} \begin{align*}\angle P\end{align*} is supplementary to \begin{align*}\angle O\end{align*} \begin{align*}\angle L \cong \angle O\end{align*} Prove: \begin{align*}\angle P \cong \angle M\end{align*}
Statement Reason
1.
2. \begin{align*}m\angle L = m\angle O \end{align*}
3. Definition of supplementary angles
4. Substitution
5. Substitution
6. Subtraction \begin{align*}PoE\end{align*}
7. \begin{align*}\angle M \cong \angle P \end{align*}
  1. Given: \begin{align*}\angle 1 \cong \angle 4\end{align*} Prove: \begin{align*}\angle 2 \cong \angle 3\end{align*}
Statement Reason
1.
2. \begin{align*}m\angle 1 = m\angle 4 \end{align*}
3. Definition of a Linear Pair

4. \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 2\end{align*} are supplementary

\begin{align*}\angle 3\end{align*} and \begin{align*}\angle 4\end{align*} are supplementary

5. Definition of supplementary angles
6. \begin{align*}m\angle 1 + m\angle 2 = m\angle 3 + m\angle 4 \end{align*}
7. \begin{align*}m\angle 1 + m\angle 2 = m\angle 3 + m\angle 1 \end{align*}
8. \begin{align*}m\angle 2 = m\angle 3 \end{align*}
9. \begin{align*}\angle 2 \cong \angle 3 \end{align*}
  1. Given: \begin{align*}\angle C\end{align*} and \begin{align*}\angle F\end{align*} are right angles Prove: \begin{align*}m\angle C + m\angle F = 180^\circ\end{align*}
Statement Reason
1.
2. \begin{align*}m\angle C = 90^\circ, m\angle F = 90^\circ \end{align*}
3. \begin{align*}90^\circ + 90^\circ = 180^\circ \end{align*}
4. \begin{align*}m\angle C + m\angle F = 180^\circ \end{align*}
  1. Given: \begin{align*}l \bot m\end{align*} Prove: \begin{align*}\angle 1\cong \angle 2\end{align*}
Statement Reason
1. \begin{align*}l \bot m\end{align*}
2. \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 2\end{align*} are right angles
3.
  1. Given: \begin{align*}m\angle 1 = 90^\circ\end{align*} Prove: \begin{align*}m\angle 2 = 90^\circ\end{align*}
Statement Reason
1.
2. \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 2\end{align*} are a linear pair
3. Linear Pair Postulate
4. Definition of supplementary angles
5. Substitution
6. \begin{align*}m\angle 2 = 90^\circ \end{align*}
  1. Given: \begin{align*}l \bot m\end{align*} Prove: \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 2\end{align*} are complements
Statement Reason
1.
2. \begin{align*}\bot\end{align*} lines create right angles
3. \begin{align*}m\angle 1 + m\angle 2 = 90^\circ \end{align*}
4. \begin{align*}\angle 1\end{align*} and \begin{align*}\angle 2\end{align*} are complementary
  1. Given: \begin{align*}l \bot m\end{align*} and \begin{align*}\angle 2 \cong \angle 6\end{align*} Prove: \begin{align*}\angle 6 \cong \angle 5\end{align*}
Statement Reason
1.
2. \begin{align*}m\angle 2 = m\angle 6 \end{align*}
3. \begin{align*}\angle 5 \cong \angle 2\end{align*}
4. \begin{align*}m\angle 5 = m\angle 2\end{align*}
5. \begin{align*}m\angle 5 = m\angle 6\end{align*}

Use the picture for questions 11-21.

Given: \begin{align*}H\end{align*} is the midpoint of \begin{align*}\overline{AE}, \overline{MP}\end{align*} and \begin{align*}\overline{GC}\end{align*}

\begin{align*}M\end{align*} is the midpoint of \begin{align*}\overline{GA}\end{align*}

\begin{align*}P\end{align*} is the midpoint of \begin{align*}\overline{CE}\end{align*}

\begin{align*}\overline{AE} \cong \overline{GC}\end{align*}

  1. List two pairs of vertical angles.
  2. List all the pairs of congruent segments.
  3. List two linear pairs that do not have \begin{align*}H\end{align*} as the vertex.
  4. List a right angle.
  5. List a pair of adjacent angles that are NOT a linear pair (do not add up to \begin{align*}180^\circ\end{align*}).
  6. What line segment is the perpendicular bisector of \begin{align*}\overline{AE}\end{align*}?
  7. Name a bisector of \begin{align*}\overline{MP}\end{align*}.
  8. List a pair of complementary angles.
  9. If \begin{align*}\overline{GC}\end{align*} is an angle bisector of \begin{align*}\angle AGE\end{align*}, what two angles are congruent?
  10. Find \begin{align*}m\angle GHE\end{align*}.

For questions 21-25, find the measure of the lettered angles in the picture below.

  1. a
  2. b
  3. c
  4. d
  5. e (hint: \begin{align*}e\end{align*} is complementary with \begin{align*}b\end{align*})

Review Queue Answers

1.

Statement Reason

1. \begin{align*}\overline{VX}\end{align*} is an \begin{align*}\angle\end{align*} bisector of \begin{align*}\angle{WVY}\end{align*}

\begin{align*}\overline{VY}\end{align*} is an \begin{align*}\angle\end{align*} bisector of \begin{align*}\angle{XVZ}\end{align*}

Given

2. \begin{align*}\angle{WVX} \cong \angle {XVY}\end{align*}

\begin{align*}\angle{XVY} \cong \angle {YVZ}\end{align*}

Definition of an angle bisector
3. \begin{align*}\angle{WVX} \cong \angle {YVZ}\end{align*} Transitive Property

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CK.MAT.ENG.SE.1.Geometry-Basic.2.5