Question

consider this diagram of quadrilateral abcd, which is not drawn to scale.
which two statements must be true based on the information indicated by the diagram?
choose 2 answers:
☐ ( mangle dab = 90^circ )
☐ ( mangle abc = 90^circ )
☐ ( angle dac cong angle bca )
☐ ( angle abd cong angle cdb )

Explanation:

1. For \( m\angle DAB = 90^\circ \): The diagram shows a right angle at \( D \), and \( AB \parallel CD \), \( AD \parallel BC \) (from the arrow marks indicating parallel sides), so \( \angle DAB \) is also a right angle as it's a rectangle - like figure (with right angle at \( D \) and parallel sides).
2. For \( \angle DAC\cong\angle BCA \): Since \( AD\parallel BC \) (from the arrow - marked parallel sides), \( AC \) is a transversal. By the Alternate Interior Angles Theorem, \( \angle DAC \) and \( \angle BCA \) are alternate interior angles, so they are congruent.
3. For \( m\angle ABC = 90^\circ \): While the figure looks like a rectangle, the right angle is marked at \( D \), and we can't be sure \( \angle ABC \) is \( 90^\circ \) just from the given diagram (the diagram is not to scale, and the right angle is only marked at \( D \)).
4. For \( \angle ABD\cong\angle CDB \): There's no sufficient information from the diagram (parallel sides and right angle at \( D \)) to conclude these angles are congruent.

Answer:

a. \( m\angle DAB = 90^\circ \)
c. \( \angle DAC\cong\angle BCA \)