Geometry Quiz: Vertical Angles & Linear Pairs

In which diagram are angles 1 and 2 vertical angles?

What is the value of m?

Marcus states that angle ORP and angle LRP are a linear pair. Which best describes his statement?

Given: mEDF = 120°; mADB = (3x)°; mBDC = (2x)°. Prove: x = 24. What is the missing reason in step 3?

Which angle is a vertical angle with EFD?

Given: mADE = 60° and mCDF = (3x + 15)°. Prove: x = 15. What is the missing statement and the missing reason in step 5?

Given: mORP = 80°; mORN = (3x + 10)°. Prove: x = 30. Which statement could be used in step 2 when proving x = 30?

In the diagram, which angle is part of a linear pair and part of a vertical pair?

What is the value of x?

Which is a pair of vertical angles?

What is the value of x? (assignment)

Given: m∠ELG = 124°. Prove: x = 28. Complete the steps in the two-column proof.

Imagine two lines intersect. How can the properties of linear pairs and vertical angles help to determine the angle measures created by the intersecting lines? Explain.

Given: m∠QVR = 49°. Prove: x = 15. Complete the steps of the proof.

What is the measure of ∠DAE?

If the m∠SRW = 85°, what are the measures of ∠VRU and ∠URW? m∠VRU = __°; m∠URW = __°

Vertical Angles

Vertical angles are formed when two lines intersect, creating pairs of opposite angles that are equal.
Examples of vertical angles include AFE and BFD, and WRS and VRT.

Linear Pairs

A linear pair consists of two adjacent angles that are formed when two lines intersect and are supplementary, meaning they sum up to 180 degrees.
An example of a linear pair includes angles ∠SRT and ∠TRV, as well as ∠VRW and ∠WRS.

Angle Measurements

Specific measurements provided include m∠SRW = 95° and m∠DAE = 45°.
If m∠SRW = 85°, then m∠VRU = 80° and m∠URW = 15° based on angle relationships.

Algebraic Proofs and Calculations

To calculate unknowns:
- In the proof involving mEDF = 120° and mADB = (3x)°, x is determined to equal 24.
- In another proof where m∠ELG = 124°, completing steps with the angle addition postulate shows 2x = 56 leading to x = 28.

Reasoning in Proofs

Common reasoning includes using definitions such as the vertical angles theorem and the property of substitution for proofs involving x.
For the statement involving angles ORP and ORN being a linear pair, it can show the relationship leading to finding unknown angle measures.

Angle Relationships

Knowing the measure of an angle in a linear pair allows for determining the other angle's measure, given both sum to 180 degrees.
Vertical angles remain equal regardless of their position in the intersecting lines.

Angle Identification

EFA is identified as an angle that is both part of a linear pair and a vertical pair.
AFB is the vertical angle corresponding with angle EFD.

Correcting Misconceptions

It is incorrect to state that angle ORP and angle LRP are a linear pair if the rays are not opposite, demonstrating the need for correct definitions in geometric relationships.