Question

2.95. a laser light is pointed at a mirror as shown at right. if ∠x measures 48°, what are the measures of ∠y and ∠z? justify your reasons. homework help 2.96. are the triangles below similar? if so, write a flowchart proof that justifies your conclusion. if not, explain how you know. homework help

Explanation:

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Step1: Recall law of reflection

The angle of incidence is equal to the angle of reflection. If $\angle x$ is the angle of incidence, and assuming the normal to the mirror surface, $\angle z$ is the angle of reflection, so $\angle z=\angle x = 48^{\circ}$.

Step2: Use linear - pair property

The normal to the mirror and the mirror surface are perpendicular. $\angle x$ and $\angle y$ form a right - angle (assuming the normal and the incident - ray relationship). So $\angle y=90^{\circ}-\angle x$.
Substitute $\angle x = 48^{\circ}$ into the formula: $\angle y=90 - 48=42^{\circ}$

Step1: Identify right - angled triangles

In $\triangle ABC$ and $\triangle DEC$, $\angle B=\angle DEC = 90^{\circ}$ (given by the right - angle symbols).

Step2: Check for common angle

$\angle C$ is common to both $\triangle ABC$ and $\triangle DEC$.

Step3: Apply AA (Angle - Angle) similarity criterion

Since two angles of $\triangle ABC$ are equal to two angles of $\triangle DEC$ ( $\angle B=\angle DEC$ and $\angle C=\angle C$), by the AA similarity criterion, $\triangle ABC\sim\triangle DEC$.

Answer:

$\angle y = 42^{\circ}$, $\angle z=48^{\circ}$

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