Math Problem Statement

Prove the congruency in each

Solution

The image you've uploaded contains four geometric diagrams, each of which seems to ask for a proof of congruency between triangles based on given information. I will go through each one and provide the steps required to prove the congruency of the triangles.

Diagram 1.1:

Given: DAB=BCD\angle DAB = \angle BCD, AB\overline{AB} is common, and DA=CB\overline{DA} = \overline{CB}.

Proof:

  1. AB\overline{AB} is common (Shared side).
  2. DA=CB\overline{DA} = \overline{CB} (Given).
  3. DAB=BCD\angle DAB = \angle BCD (Given).
  4. By the SAS (Side-Angle-Side) congruence rule, DABBCD\triangle DAB \cong \triangle BCD.

Diagram 1.2:

Given: AB=CD\overline{AB} = \overline{CD}, BAE=CED\angle BAE = \angle CED, and AE=ED\overline{AE} = \overline{ED}.

Proof:

  1. AB=CD\overline{AB} = \overline{CD} (Given).
  2. BAE=CED\angle BAE = \angle CED (Given).
  3. AE=ED\overline{AE} = \overline{ED} (Given).
  4. By the ASA (Angle-Side-Angle) congruence rule, BAECED\triangle BAE \cong \triangle CED.

Diagram 1.3:

Given: AB=CD\overline{AB} = \overline{CD}, AC=BD\overline{AC} = \overline{BD}, and BAC=DAB\angle BAC = \angle DAB.

Proof:

  1. AB=CD\overline{AB} = \overline{CD} (Given).
  2. AC=BD\overline{AC} = \overline{BD} (Given).
  3. BAC=DAB\angle BAC = \angle DAB (Given).
  4. By the SAS (Side-Angle-Side) congruence rule, ABCBDA\triangle ABC \cong \triangle BDA.

Diagram 1.4:

Given: AB=CD\overline{AB} = \overline{CD}, DAB=BCD\angle DAB = \angle BCD, and AD=BC\overline{AD} = \overline{BC}.

Proof:

  1. AB=CD\overline{AB} = \overline{CD} (Given).
  2. DAB=BCD\angle DAB = \angle BCD (Given).
  3. AD=BC\overline{AD} = \overline{BC} (Given).
  4. By the ASA (Angle-Side-Angle) congruence rule, DABBCD\triangle DAB \cong \triangle BCD.

These steps establish the congruency of the triangles in each of the diagrams provided. Let me know if you need further clarification or a more detailed explanation.

Would you like any additional details or have any questions?

Here are five questions you might find useful:

  1. How can the ASA and SAS congruence rules be applied in other geometric proofs?
  2. What are other congruence rules for triangles besides ASA and SAS?
  3. How can you determine when to use which congruence rule?
  4. What is the significance of a common side in proving congruence?
  5. How does angle bisector theorem apply in triangle congruency?

Tip: When working on geometric proofs, always start by identifying the given information and the goal, then match them with the appropriate congruence rules (like SSS, SAS, ASA, or AAS).

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Math Problem Analysis

Mathematical Concepts

Geometry
Triangle Congruence

Formulas

-

Theorems

ASA (Angle-Side-Angle) congruence rule
SAS (Side-Angle-Side) congruence rule

Suitable Grade Level

High School