Geometry: Triangle Congruence and Isosceles Properties

Congruence in Triangles

Problem 11: Congruence of Triangles PQR and PRM

Given Conditions:

  • Triangle PQR is isosceles with sides |PQ| = |PR|.
  • M is the midpoint of segment [PQ]
  • N is the midpoint of segment [PR]

To Prove:
Triangle PQN is congruent to triangle PRM.

Proof Steps:

  1. Identify Sides:
       - Since M is the midpoint of [PQ], then |PM| = |MQ|.
       - Since N is the midpoint of [PR], then |PN| = |NR|.

  2. Use Isosceles Triangle Properties:
       - From triangle PQR, we know |PQ| = |PR| implies that |PM| = |NR| because both segments are half of equal sides, thus |PM| = |NR|.    

  3. Angles:
       - The angles at point P are shared between triangles PQN and PRM, hence angle P is equal for both triangles (Angle P = Angle P).    

  4. Apply the Side-Angle-Side (SAS) Congruence Rule:
       - We have two sides equal and the included angle equal:
         - |PM| = |NR| (sides)
         - Angle P (included angle)

Thus, by the SAS criterion, triangle PQN is congruent to triangle PRM.

Problem 12: Congruence of Triangles ADC and ABC

Given Conditions:

  • Quadrilateral ABCD where |AD| = |AB|
  • Angle ADC = angle ABC = 90°

To Prove:
Triangles ADC and ABC are congruent.

Proof Steps:
  1. Identify Sides:
  • From the condition |AD| = |AB|, we have one pair of equal sides between triangles ADC and ABC.
  1. Right Angles:
  • The angles provided: angle ADC and angle ABC are both right angles, i.e., 90°. Therefore,
      Angle ADC = Angle ABC = 90°.   
  1. Shared Side:
  • The segment |AC| is common to both triangles. Thus,
      |AC| = |AC| (shared side).
  1. Apply the Hypotenuse-Leg (HL) Congruence Theorem:
  • Having one leg and the hypotenuse equal provides sufficient criteria for the triangles to be congruent:   - Hypotenuse |AD| = |AB|   - Leg |AC| = |AC| (shared)

Thus, by the HL theorem, triangles ADC and ABC are congruent.

Problem 13: Properties of Isosceles Triangle

Given Conditions:

  • Triangle ABC with |AB| = |AC|.

Note:

  • The properties of isosceles triangles imply that angles opposite the equal sides are also equal.
  • Therefore, angle B = angle C.

This fundamental property plays an important role in proving congruence in various configurations involving isosceles triangles.

Concluding Points
  • The concept of congruence in triangles is pivotal in geometry, particularly relating to proving equality of different triangle configurations through various theorems such as SAS, ASA, and HL.
  • Understanding these properties enables the application of geometric principles to solve problems and prove relationships in triangles and larger geometric figures.