Question

1. consider the diagram.

a. if the angle measures in $\triangle ghi$ and $\triangle jkl$ can be found, list them in the table. then identify the angle relationship that allowed you to find the measure.

angle in $\triangle ghi$	corresponding angle in $\triangle jkl$	angle relationship

b. is $\triangle ghi$ similar to $\triangle jkl$ by the angle - angle criterion? explain.

Explanation:

Step1: Identify right angles

Both $\triangle GHI$ and $\triangle JKL$ have a right angle (marked by the square symbol), so $\angle GHI = 90^\circ$ and $\angle JKL = 90^\circ$. These are corresponding right angles from parallel vertical lines.

Step2: Identify congruent acute angles

$\angle GIH = 34^\circ$. Since the slanted line is a transversal cutting parallel vertical lines, corresponding angles are congruent, so $\angle JLK = 34^\circ$.

Step3: Calculate third angles

For $\triangle GHI$: $\angle HGI = 180^\circ - 90^\circ - 34^\circ = 56^\circ$.
For $\triangle JKL$: $\angle KJL = 180^\circ - 90^\circ - 34^\circ = 56^\circ$. These are corresponding congruent angles.

Step4: Verify AA similarity

Two pairs of corresponding angles are congruent ($90^\circ$ and $34^\circ$), so the triangles satisfy the angle-angle (AA) similarity criterion.

Part a Table:

Angle in $\triangle GHI$	Corresponding Angle in $\triangle JKL$	Angle Relationship
$\angle GIH = 34^\circ$	$\angle JLK = 34^\circ$	Corresponding Angles (Transversal on Parallel Lines)
$\angle HGI = 56^\circ$	$\angle KJL = 56^\circ$	Corresponding Angles (Derived from Triangle Angle Sum)

Answer:

Part a:

See the completed table above.

Part b:

Yes, $\triangle GHI$ is similar to $\triangle JKL$ by the angle-angle criterion. Both triangles have two pairs of congruent corresponding angles: a pair of $90^\circ$ right angles, and a pair of $34^\circ$ acute angles. When two pairs of corresponding angles of two triangles are congruent, the triangles are similar by the AA similarity postulate.