Question

4.6 exercises
developing proof for exercises 1–9, copy the figures onto your paper and mark them with the given information. answer the question about segment or angle congruence. if your answer is yes, write a paragraph proof explaining why. remember to use your reasoning strategies, especially to apply previous conjectures, and add an auxiliary line. if there is not enough information to prove congruence, write “cannot be determined.” otherwise state which congruence shortcut you used.

1. ∠a ≅ ∠c, ∠abd ≅ ∠cbd is ( overline{ab} cong overline{cb} )?
2. ( overline{cn} cong overline{wn} ), ∠c ≅ ∠w, ( overline{rn} cong overline{on} )
3. ( overline{cs} cong overline{hr} ), ∠1 ≅ ∠2 is ( overline{cr} cong overline{hs} )?
4. ∠s ≅ ∠i, ∠g ≅ ∠a, t is the midpoint of ( overline{si} ). is ( overline{sg} cong overline{ir} )?
5. ( overline{fi} cong overline{fe} ), ( overline{vi} cong overline{ve} ) is ∠i ≅ ∠e?
6. ( overline{mn} cong overline{ma} ), ( overline{me} cong overline{mr} ) is ∠e ≅ ∠r?
7. ( overline{bt} cong overline{eu} ), ( overline{bu} cong overline{et} ) is ∠b ≅ ∠e?
8. half is a parallelogram. is ( overline{ha} cong overline{hf} )?
9. ∠d ≅ ∠c, ∠o ≅ ∠a, ∠g ≅ ∠t. is ( overline{ta} cong overline{go} )?

you will need

• construction tools (for exercise 17) and 18
• geometry software (for exercise 23)

Explanation:

Step1: Analyze Given Info

We have \( \angle A \cong \angle C \), \( \angle ABD \cong \angle CBD \), and \( BD \) is common to \( \triangle ABD \) and \( \triangle CBD \).

Step2: Apply AAS Congruence

In \( \triangle ABD \) and \( \triangle CBD \):

• \( \angle A \cong \angle C \) (given)
• \( \angle ABD \cong \angle CBD \) (given)
• \( BD \cong BD \) (reflexive property)

By the Angle - Angle - Side (AAS) congruence criterion, \( \triangle ABD \cong \triangle CBD \).

Step3: Corresponding Parts of Congruent Triangles

Since \( \triangle ABD \cong \triangle CBD \), their corresponding sides \( AB \) and \( CB \) are congruent. So, \( \overline{AB} \cong \overline{CB} \).

Answer:

Yes, \( \overline{AB} \cong \overline{CB} \) by AAS congruence ( \( \triangle ABD \cong \triangle CBD \) as \( \angle A \cong \angle C \), \( \angle ABD \cong \angle CBD \), \( BD \cong BD \) ).