Question

in the diagram, line x is parallel to line y, m∠1 = 65°, and m∠7 = 55°. stuart says that m∠12 = 60°. his reasoning is shown. step 1: m∠8 = 60°, because m∠1 + m∠7 + m∠8 = 180°. step 2: ∠8 ≅ ∠12, because ∠8 and ∠12 are corresponding angles. step 3: so, m∠12 = 60°. use the drop - down menus to explain whether or not stuart is correct. click the arrows to choose an answer from each menu. the sum of ∠1, ∠7 and ∠8 is choose... ∠8 and ∠12 are choose... the measure of ∠12 must be choose... stuart is choose...

Explanation:

Step1: Verify Step 1 logic

The sum of angles in a triangle is $180^\circ$, so $m\angle1 + m\angle7 + m\angle8 = 180^\circ$. Substitute given values:
$$65^\circ + 55^\circ + m\angle8 = 180^\circ$$
$$120^\circ + m\angle8 = 180^\circ$$
$$m\angle8 = 180^\circ - 120^\circ = 60^\circ$$
This step is correct.

Step2: Verify Step 2 logic

$\angle8$ and $\angle12$ are alternate interior angles (not corresponding angles) because lines $x \parallel y$ and the transversal creates them inside the parallel lines, on opposite sides of the transversal. While $\angle8 \cong \angle12$ is true, the reasoning for why is wrong.

Step3: Evaluate overall correctness

Stuart's final measure for $\angle12$ is numerically correct, but his justification for Step 2 is incorrect.

Answer:

• The sum of $\angle1$, $\angle7$ and $\angle8$ is $\boldsymbol{180^\circ}$
• $\angle8$ and $\angle12$ are $\boldsymbol{\text{alternate interior angles}}$ (not corresponding angles)
• The measure of $\angle12$ must be $\boldsymbol{60^\circ}$
• Stuart is $\boldsymbol{\text{incorrect}}$ (his reasoning for Step 2 is wrong, even though the final angle measure matches)