Question

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Answer 1

a) SAS ≅
ΔABD ≅ ΔCBD

b) AA ~
ΔABE ~ ΔDCE

c) AAS ≅
ΔABC ≅ ΔDEF

Explanation:

Part a

Side-Angle-Side (SAS) Congruence Theorem

If two sides and the included angle in one triangle are congruent to two sides and the included angle in another triangle, the triangles are congruent.

From inspection of the given diagram:

• The two triangles ABD and CBD share side BD.
• The tick marks on sides BA and BC indicate they are the same length.
• The tick marks on angles ABD and CBD indicate they are the same measure.

Therefore, as two sides and the included angle of both triangles are congruent, the triangles are congruent by SAS congruence theorem.

• SAS ≅
• ΔABD ≅ ΔCBD

`\hrulefill`

Part b

Angle-Angle Similarity

If any two angles of a triangle are equal to any two angles of another triangle, then the two triangles are similar to each other.

As line segment AB is parallel to line segment CD:

• According to the Vertical Angles Theorem, m∠BEA ≅ m∠CED.
• According to the Alternate Interior Angles Theorem, m∠ABE ≅ m∠DCE and m∠BAE ≅ m∠CDE.

Therefore, as the corresponding interior angles of the triangles are congruent, according to the Angle-Angle Similarity Theorem, the two triangles are similar to each other:

• AA ~
• ΔABE ~ ΔDCE

Note: We cannot say the triangles are congruent, since there is no indication that any of the sides are equal in length on the given diagram.

`\hrulefill`

Part c

Angle-Angle-Side (AAS) Congruence Theorem

If any two angles and the non-included side of one triangle are congruent to the corresponding parts of another triangle, the triangles are congruent.

The interior angles of a triangle sum to 180°.

Therefore, angle B of triangle ABC is 45°, and angle D of triangle DEF is 73°. So the interior angles of triangles ABC and DEF are the congruent, which means the triangles are similar.

If the triangles are congruent, we would expect one of their corresponding sides to be equal in length. So AB = DE, or BC = EF, or AC = DF.

From inspection of the triangles, we can see that sides BC and EF are both 13 units in length. Therefore, BC = EF.

Therefore, as two angles and the non-included side of both triangles are congruent, the triangles are congruent by AAS congruence theorem.

• AAS ≅
• ΔABC ≅ ΔDEF

Answer 2

a. congruent triangle

b. similar triangle

c. congruent triangle

Explanation:

a.

In Δ ABD and ΔCBD

BD=BD reflexive property

∡ABD=∡CBD given

AB=BC given

Δ ABD ≅ ΔCBD by SAS (Side-Angle-Side) axiom.

b.

In ΔABE and ΔDCE

∡ABE=∡ DCE being alternative angle

∡BAE=∡CDE being alternative angle

∡AEB=∡CED

Δ ABE
`\bold{\sim}` ΔCBD by AAA (Angle-Angle-Angle) axiom.

Since, all the corresponding angle are equal. So, it is similar triangle not congruent.

C.

In ΔABC and ΔDEF

∡F=∡C=62° given

∡D=∡B =45° given
FE=BC given

Δ ABC ≅ ΔDEF by Angle-Angle-Side axiom.