Question

1. given the figure shown, complete the flowchart proof to prove that $\triangle abc \sim \triangle def$.

Explanation:

Step1: Recall Triangle Angle Sum

The sum of angles in a triangle is \(180^\circ\), so for \(\triangle ABC\), \(m\angle A + m\angle B + m\angle C = 180^\circ\) (this is the triangle angle - sum theorem).

Step2: Substitute Known Angles in \(\triangle ABC\)

We know that \(m\angle A = 45^\circ\) and \(m\angle C=104^\circ\). Substitute these values into the angle - sum formula: \(45^\circ+m\angle B + 104^\circ=180^\circ\).

Step3: Solve for \(m\angle B\)

First, simplify the left - hand side: \(45 + 104+m\angle B=149 + m\angle B\). Then, solve the equation \(149 + m\angle B=180\) for \(m\angle B\). Subtract 149 from both sides: \(m\angle B=180 - 149=31^\circ\).

Step4: Identify Corresponding Angles

We can see that \(m\angle A = m\angle D = 45^\circ\) (given, \(\angle A\) and \(\angle D\) are both \(45^\circ\)) and \(m\angle B=m\angle E = 31^\circ\) (we just calculated \(m\angle B = 31^\circ\) and \(\angle E\) is \(31^\circ\)).

Step5: Apply AA (Angle - Angle) Similarity Criterion

If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar (AA similarity criterion). Since \(\angle A\cong\angle D\) and \(\angle B\cong\angle E\), \(\triangle ABC\sim\triangle DEF\) by AA similarity.

For the flowchart:

1. The first box (label 1) should be "Triangle Angle - Sum Theorem" (justifying \(m\angle A + m\angle B + m\angle C = 180^\circ\)).
2. The second box (label 2) is the substitution step: "Substitute \(m\angle A = 45^\circ\), \(m\angle C = 104^\circ\)".
3. The third box (label 3) is the calculation: "Solve \(45 + m\angle B+104 = 180\) for \(m\angle B\)".
4. The fourth box (label 4) is "Given \(m\angle D = 45^\circ\)" (to show \(\angle A\cong\angle D\)).
5. The fifth box (label 5) is "Corresponding angles are congruent" (to show \(\angle B\cong\angle E\) since \(m\angle B = m\angle E=31^\circ\)).
6. The sixth box (label 6) is "AA Similarity Criterion" (justifying \(\triangle ABC\sim\triangle DEF\)).

Answer:

The flowchart is completed by using the triangle angle - sum theorem, substituting known angles, solving for the unknown angle, identifying congruent corresponding angles, and applying the AA similarity criterion. The key steps show that \(\triangle ABC\sim\triangle DEF\) by AA similarity because \(\angle A\cong\angle D\) and \(\angle B\cong\angle E\).