USING CONGRUENT TRIANGLES WORKSHEET

Problem 1 :

In the diagram shown below, AB || CD, BC || DA. Prove that

AB  ≅  CD

Problem 2 :

In the diagram shown below,

A is the midpoint of MT

A is the midpoint of SR

Prove that MS || TR.

Problem 3 :

In the diagram shown below,

∠1  ≅  ∠2

∠3  ≅  ∠4

Prove that ΔBCE  ≅  ΔDCE.

Problem 1 :

In the diagram shown below, AB || CD, BC || DA. Prove that

AB  ≅  CD

Plane for Proof :

Show that ΔABD  ≅  ΔCDB. Then use the fact that corresponding parts of congruent triangles are congruent.

Solution :

First copy the diagram and mark it with the given information. Then mark any additional information that we can deduce. Because AB and CD are parallel segments intersected by a transversal, BC and DA are parallel segments intersected by a transversal, we can deduce that two pairs of alternate interior angles are congruent.

Mark Given Information :

Paragraph Proof :

Because AB || CD, it follows from the Alternate Interior Angles Theorem, that  ∠ABD  ≅  ∠CDB. For the same reason, ∠ADB  ≅  ∠CBD, because BC || DA. By the Reflexive Property of Congruence, BD  ≅  BD. We can use the ASA congruence postulate to conclude that

ΔABD  ≅  ΔCDB.

Finally, because corresponding parts of congruent triangles are congruent, it follows that

AB  ≅  CD

Problem 2 :

In the diagram shown below,

A is the midpoint of MT

A is the midpoint of SR

Prove that MS || TR.

Plane for Proof :

Prove that ΔMAS  ≅  ΔTAR. Then use the fact that corresponding parts of congruent triangles are congruent to show that ∠M  ≅  ∠T. Because these angles are formed by two segments intersected by a transversal, we can conclude that MS || TR.

Solution :

 StatementsA is the midpoint of MT A is the midpoint of SRMA ≅ TA, SA ≅ RA∠MAS ≅ ∠TARΔMAS  ≅  ΔTARaaaaaaaa∠M ≅ ∠Taaaaaaa aaaaaaaaaaaaaaaaaaaaaaa aaaaaaaaaaaaaaaaaaaaaaaMS || TR ReasonsGivenGivenDefinition of midpointVertical Angles TheoremSAS Congruence PostulateCorresponding parts of congruent triangles are congruent. Alternate Interior Angles Converse

Problem 3 :

In the diagram shown below,

∠1  ≅  ∠2

∠3  ≅  ∠4

Prove that ΔBCE  ≅  ΔDCE.

Plane for Proof :

The only information we have about ΔBCE and ΔDCE is that ∠1 ≅ ∠2 and that CE ≅ CE. Notice, however, that sides BC and DC are also sides of ΔABC and ΔADC. If we can prove that ΔABC ≅ ΔADC, we can use the fact that corresponding parts of congruent triangles are congruent to get a third piece of information about ΔBCE and ΔDCE.

Solution :

 Statements∠1  ≅  ∠2∠3  ≅  ∠4AC  ≅  ACΔABC  ≅  ΔADCBC  ≅  DCCE  ≅  CEΔBCE  ≅  ΔDCE ReasonsGivenGivenReflexive Property of CongruenceASA Congruence PostulateCorresponding parts of ≅ Δ are ≅Reflexive Property of CongruenceSAS Congruence Postulate

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