Question

geometry homework #5
due: 9/19
triangle congruence

1. a reflection by line lm takes triangle lmn onto triangle lmp.

a. explain why the image of segment mn coincides with segment mp.
b. explain why the image of angle n coincides with angle p.
c. is triangle lmn congruent to triangle lmp? explain your reasoning.

1. sketch the unique triangles that can be made with two side lengths of 7 and the included angle measuring 60 degrees. how do you know you have sketched all possibilities?

Explanation:

Step1: Understand reflection property

A reflection is a rigid - motion. Points on the line of reflection are invariant. For a point \(N\) and its image \(P\) under reflection over line \(LM\), the line \(LM\) is the perpendicular bisector of the segment \(NP\). Since \(M\) lies on the line of reflection \(LM\), and reflection preserves distances, the distance from \(M\) to \(N\) is equal to the distance from \(M\) to \(P\). So, the image of segment \(MN\) coincides with segment \(MP\).

Step2: Angle - reflection property

Reflection is a rigid - motion that preserves angle measures. The angle \(N\) and its image angle \(P\) are corresponding angles under the reflection over line \(LM\). Since rigid - motions preserve angle measures and the orientation of the angles with respect to the line of reflection, the image of angle \(N\) coincides with angle \(P\).

Step3: Prove triangle congruence

A reflection is a rigid - motion. Rigid - motions preserve side lengths and angle measures. In \(\triangle LMN\) and \(\triangle LMP\), \(LM = LM\) (common side), \(MN=MP\) (from part a, property of reflection), and \(\angle N=\angle P\) (from part b, property of reflection). By the Side - Angle - Side (SAS) congruence criterion, \(\triangle LMN\cong\triangle LMP\).

Step4: Sketch unique triangle

By the Side - Angle - Side (SAS) congruence criterion, if two sides and the included angle of a triangle are given, the triangle is unique. To sketch the triangle: First, draw a ray. Mark a point on the ray. Using a compass, set the width to 7 units and draw an arc from the marked point. Then, using a protractor, measure an angle of \(60^{\circ}\) from the ray at the marked point. Set the compass width to 7 units again and draw an arc from the end - point of the first arc that intersects the ray of the \(60^{\circ}\) angle. Connect the points to form the triangle. There is only one such triangle because the SAS congruence criterion states that given two side lengths and the measure of the included angle, there is exactly one triangle that can be formed.

Answer:

a. Because reflection over line \(LM\) preserves distances and \(M\) is on the line of reflection, so \(MN = MP\) and the segments coincide.
b. Because reflection preserves angle measures and \(N\) and \(P\) are corresponding angles under reflection over \(LM\), the angles coincide.
c. Yes, \(\triangle LMN\cong\triangle LMP\) by the SAS congruence criterion since \(LM = LM\) (common side), \(MN = MP\) (from reflection property), and \(\angle N=\angle P\) (from reflection property).

1. There is 1 unique triangle. By the SAS congruence criterion, given two side lengths of 7 and an included angle of \(60^{\circ}\), only one triangle can be formed.