Question

Given start overline, b, c, end overline, parallel, start overline, a, d, end overline bc ∥ ad , complete the flowchart proof below. a b c d start overline, b, c, end overline bc parallel∥ start overline, a, d, end overline ad reason: given ↓ angle, b, c, a∠bca cong≅ angle, d, a, c∠dac reason: angle, b∠b cong≅ angle, d∠d reason: start overline, a, c, end overline ac cong≅ start overline, a, c, end overline ac reason: †˜↓†™ triangle, a, b, c, cong, triangle, c, d, aΔabc≅Δcda reason:

Answer 1

The flowchart proof demonstrates that, given start
`\( \overline{b, c} \)` is parallel to
`\( \overline{a, d} \)`, we can conclude that
`\( \angle bca \)` is congruent to
`\( \angle dac \)`, and
`\( \angle b \)` is congruent to
`\( \angle d \)`. This is supported by the fact that
`\( \overline{a, c} \)`is congruent to
`\( \overline{a, c} \)`, and triangles
`\( \triangle abc \)` and
`\( \triangle cda \)` are congruent.

Step-by-step explanation:

The given information states that
`\( \overline{b, c} \)`is parallel to
`\( \overline{a, d} \) (BC \( \parallel \) AD).` From this, we can deduce that alternate interior angles are congruent. Therefore,
`\( \angle bca \)`is congruent to
`\( \angle dac \).`This is the first key step in establishing the congruence between the two triangles.

The second piece of information is that
`\( \angle b \)` is congruent to
`\( \angle d \).`This is a consequence of the fact that corresponding angles formed by the parallel lines
`\( \overline{b, c} \)` and
`\( \overline{a, d} \)` are congruent. This establishes the second angle congruence.

Now, with the two angles and a side
`(\( \overline{a, c} \))`being congruent, we can apply the Angle-Side-Angle (ASA) congruence criterion to prove that
`\( \triangle abc \)`is congruent to
`\( \triangle cda \).` This is due to the fact that a triangle is congruent to another if they have two corresponding angles and the included side congruent. Therefore, the triangles
`\( \triangle abc \)` and
`\( \triangle cda \)` are congruent, and the proof is complete.