Question

1. write four true statements about the angles of this figure.

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1. lines cb and de meet at point a. if angle cae measures 20°, then complete the following statements.

• angle dac measures ______ degrees
• angle bae measures ______ degrees
• angle cae is equal to angle ______

Explanation:

Step1: Analyze the first figure

• Vertical angles: Vertical angles are equal. So, \(y = x\) (since \(\angle EBD\) and \(\angle ABC\) are vertical angles).
• Linear pairs: Angles on a straight - line sum to \(180^{\circ}\). So, \(y+z = 180\) (since \(\angle EBD\) and \(\angle DBA\) form a linear pair).
• Another linear pair: \(x + w=180\) (since \(\angle ABC\) and \(\angle CBE\) form a linear pair).
• Vertical angles again: \(z = w\) (since \(\angle DBA\) and \(\angle CBE\) are vertical angles).

Step2: Analyze the second figure

• Linear pair for \(\angle DAC\):
• \(\angle CAE\) and \(\angle DAC\) form a linear pair. Since \(\angle CAE = 20^{\circ}\), and \( \angle CAE+\angle DAC=180^{\circ}\) (linear pair of angles), then \(\angle DAC=180 - 20=160^{\circ}\).
• Vertical angles for \(\angle BAE\):
• \(\angle CAE\) and \(\angle BAD\) are vertical angles. So \(\angle BAD = 20^{\circ}\). \(\angle BAE\) and \(\angle BAD\) form a linear pair. Let \(\angle BAE=x\), then \(x + 20=180\), so \(x = 160^{\circ}\).
• Vertical angles for the last blank:
• \(\angle CAE\) and \(\angle BAD\) are vertical angles. So \(\angle CAE=\angle BAD\).

Answer:

4)

• \(y = x\)
• \(y + z=180\)
• \(x + w = 180\)
• \(z = w\)

5)

• angle \(DAC\) measures \(160\) degrees
• angle \(BAE\) measures \(160\) degrees
• angle \(CAE\) is equal to angle \(BAD\)