Given that parallelogram JKLM is a…

Maths Questions
Short Answer
Step by Step Solution
Mathematics Questions

Given that parallelogram JKLM is a rectangle, it follows that ∠JML and ∠KLM are right angles. Show that ∠JML ≅ ∠KLM, and justify this using the congruence of sides JM and KL. Additionally, explain how ML is congruent to itself by the reflexive property, and conclude that triangles △JML and △KLM are congruent. Lastly, confirm that JL is congruent to MK.

Short Answer

The answer explains the properties of a rectangle, focusing on right angles and congruent sides in a parallelogram. It establishes side congruence and applies the SAS congruence postulate to conclude that two triangles within the rectangle are congruent, specifically stating that the corresponding parts of the triangles are congruent.

Step-by-Step Solution

Step 1: Understand the Properties of a Rectangle

A rectangle, such as parallelogram JKLM, is defined by having all four interior angles as right angles. This implies that the angles ∠JML and ∠KLM are both right angles. According to the definition of right angles, they are all congruent.

Step 2: Establish Side Congruence in the Parallelogram

In any parallelogram, including rectangles, opposite sides are congruent. This means that in parallelogram JKLM, we can assert that:

  • JM ‚âÖ KL
  • ML ‚âÖ JK (by reflexive property)

These congruent sides will play a crucial role in proving triangle congruence later on.

Step 3: Apply the SAS Congruence Postulate

With the angle and side congruences established, we can use the SAS congruence postulate. This postulate states that if two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, then the two triangles are congruent. Thus, we have:

  • Since ‚à†JML ‚âÖ ‚à†KLM
  • JM ‚âÖ KL, and ML ‚âÖ JK

Therefore, ΔJML ≅ ΔKLM, leading to the conclusion that corresponding parts are congruent, specifically JL ≅ MK.

Related Concepts

Rectangle

Defining shape with opposite sides equal and all angles right angles

Congruence

The property of figures or shapes being equal in size and shape

Sas congruence postulate

A rule stating that if two sides and the included angle of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.

Table Of Contents
  1. Given that parallelogram JKLM is a rectangle, it follows that ∠JML and ∠KLM are right angles. Show that ∠JML ≅ ∠KLM, and justify this using the congruence of sides JM and KL. Additionally, explain how ML is congruent to itself by the reflexive property, and conclude that triangles △JML and △KLM are congruent. Lastly, confirm that JL is congruent to MK.

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