QUESTION IMAGE
Question
- straight angles are extremely important in geometry. when two lines intersect, they form four angles, each with a common vertex. in the diagram below, lines $overline{gm}$ and $overline{fh}$ intersect at $e$. if any one of the four angles formed is given, the other three can be found quickly. fill in the table below.
| $mangle gef$ | $mangle fem$ | $mangle meh$ | $mangle heg$ |
|---|---|---|---|
| $150^{circ}$ | |||
| $25^{circ}$ | |||
| $45^{circ}$ |
Step1: Recall vertical - angle and linear - pair properties
Vertical angles are equal and linear - pair angles are supplementary (sum to 180°).
Step2: First row
If \(m\angle GEF = 120^{\circ}\), then \(m\angle MEH=m\angle GEF = 120^{\circ}\) (vertical angles). Also, \(m\angle FEM = 180 - 120=60^{\circ}\) (linear - pair with \(\angle GEF\)) and \(m\angle HEG=m\angle FEM = 60^{\circ}\) (vertical angles).
Step3: Second row
If \(m\angle MEH = 150^{\circ}\), then \(m\angle GEF=m\angle MEH = 150^{\circ}\) (vertical angles). Also, \(m\angle FEM = 180 - 150 = 30^{\circ}\) (linear - pair with \(\angle MEH\)) and \(m\angle HEG=m\angle FEM = 30^{\circ}\) (vertical angles).
Step4: Third row
If \(m\angle FEM = 25^{\circ}\), then \(m\angle HEG=m\angle FEM = 25^{\circ}\) (vertical angles). Also, \(m\angle GEF = 180 - 25=155^{\circ}\) (linear - pair with \(\angle FEM\)) and \(m\angle MEH=m\angle GEF = 155^{\circ}\) (vertical angles).
Step5: Fourth row
If \(m\angle HEG = 45^{\circ}\), then \(m\angle FEM=m\angle HEG = 45^{\circ}\) (vertical angles). Also, \(m\angle GEF = 180 - 45 = 135^{\circ}\) (linear - pair with \(\angle HEG\)) and \(m\angle MEH=m\angle GEF = 135^{\circ}\) (vertical angles).
| \(m\angle GEF\) | \(m\angle FEM\) | \(m\angle MEH\) | \(m\angle HEG\) |
|---|---|---|---|
| \(150^{\circ}\) | \(30^{\circ}\) | \(150^{\circ}\) | \(30^{\circ}\) |
| \(155^{\circ}\) | \(25^{\circ}\) | \(155^{\circ}\) | \(25^{\circ}\) |
| \(135^{\circ}\) | \(45^{\circ}\) | \(135^{\circ}\) | \(45^{\circ}\) |
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Step1: Recall vertical - angle and linear - pair properties
Vertical angles are equal and linear - pair angles are supplementary (sum to 180°).
Step2: First row
If \(m\angle GEF = 120^{\circ}\), then \(m\angle MEH=m\angle GEF = 120^{\circ}\) (vertical angles). Also, \(m\angle FEM = 180 - 120=60^{\circ}\) (linear - pair with \(\angle GEF\)) and \(m\angle HEG=m\angle FEM = 60^{\circ}\) (vertical angles).
Step3: Second row
If \(m\angle MEH = 150^{\circ}\), then \(m\angle GEF=m\angle MEH = 150^{\circ}\) (vertical angles). Also, \(m\angle FEM = 180 - 150 = 30^{\circ}\) (linear - pair with \(\angle MEH\)) and \(m\angle HEG=m\angle FEM = 30^{\circ}\) (vertical angles).
Step4: Third row
If \(m\angle FEM = 25^{\circ}\), then \(m\angle HEG=m\angle FEM = 25^{\circ}\) (vertical angles). Also, \(m\angle GEF = 180 - 25=155^{\circ}\) (linear - pair with \(\angle FEM\)) and \(m\angle MEH=m\angle GEF = 155^{\circ}\) (vertical angles).
Step5: Fourth row
If \(m\angle HEG = 45^{\circ}\), then \(m\angle FEM=m\angle HEG = 45^{\circ}\) (vertical angles). Also, \(m\angle GEF = 180 - 45 = 135^{\circ}\) (linear - pair with \(\angle HEG\)) and \(m\angle MEH=m\angle GEF = 135^{\circ}\) (vertical angles).
| \(m\angle GEF\) | \(m\angle FEM\) | \(m\angle MEH\) | \(m\angle HEG\) |
|---|---|---|---|
| \(150^{\circ}\) | \(30^{\circ}\) | \(150^{\circ}\) | \(30^{\circ}\) |
| \(155^{\circ}\) | \(25^{\circ}\) | \(155^{\circ}\) | \(25^{\circ}\) |
| \(135^{\circ}\) | \(45^{\circ}\) | \(135^{\circ}\) | \(45^{\circ}\) |