Question

complete the following problems. show all your work.

learning goal from lesson 6.2, 6.3, and 7.3
i can prove theorems about geometric figures using triangle congruence and similarity.

1. identify the sides or angles that need to be congruent in order to make the given triangles congruent by aas. (lesson 6.2) (1 point)

a.
b.

1. which of the following properties prove that the triangles are congruent? (lesson 6.2 and 6.3) (1 point)

a.
a asa congruence theorem
b aas congruence theorem
c sas congruence theorem
d hl congruence theorem

b. which statement would prove \\(\delta amc \cong \delta bmc\\) by hl congruence theorem?
a \\(\angle a \cong \angle b\\)
b \\(\angle mca \cong \angle mcb\\)
c \\(\angle c \cong \angle c\\)
d \\(mc \cong mc\\)

Explanation:

Analyze Question 1a for AAS Congruence

Given: \(\angle B \cong \angle E\) and \(\angle C \cong \angle F\).
For Angle-Angle-Side (AAS) congruence, we need a non-included side to be congruent.
The included side between the given angles is \(BC\) and \(EF\).
Therefore, the non-included corresponding sides must be congruent:

$$ \overline{AC} \cong \overline{DF} \quad \text{or} \quad \overline{AB} \cong \overline{DE} $$

Analyze Question 1b for AAS Congruence

Given: \(\angle A \cong \angle D\) and \(\overline{BC} \cong \overline{EF}\).
For AAS congruence, the congruent side must be non-included.
The side \(BC\) is opposite to \(\angle A\), and \(EF\) is opposite to \(\angle D\).
We need a second pair of corresponding angles such that the side remains non-included.
The second angle must be:

$$ \angle C \cong \angle F $$

Analyze Question 2a for Congruence Theorem

Given markings:

• One pair of congruent angles: \(\angle B \cong \angle E\)
• A second pair of congruent angles: \(\angle C \cong \angle F\)
• One pair of congruent sides: \(\overline{AC} \cong \overline{DF}\)

The side \(\overline{AC}\) is opposite to \(\angle B\), and \(\overline{DF}\) is opposite to \(\angle E\) (non-included).
This matches the Angle-Angle-Side (AAS) Congruence Theorem.

Analyze Question 2b for HL Congruence

Given: \(\overline{AC} \cong \overline{BC}\) (hypotenuses of right triangles \(\triangle AMC\) and \(\triangle BMC\)).
The triangles share a common leg: \(\overline{MC} \cong \overline{MC}\) by the Reflexive Property.
To use the Hypotenuse-Leg (HL) Congruence Theorem, we must establish that the triangles are right triangles.
This requires \(\overline{CM} \perp \overline{AB}\), which means \(\angle AMC \cong \angle BMC = 90^\circ\).
Thus, the statement needed is:

$$ \angle MCA \cong \angle MCB \quad \text{is incorrect, we need } \angle AMC \cong \angle BMC \text{ or perpendicularity.} $$

Looking at the options:

• A: \(\angle A \cong \angle B\)
• B: \(\angle MCA \cong \angle MCB\)
• C: \(\angle C \cong \angle C\)
• D: \(\overline{MC} \cong \overline{MC}\) (Reflexive property providing the congruent leg).

Answer:

Question 1a

\(\overline{AC} \cong \overline{DF}\) (or \(\overline{AB} \cong \overline{DE}\))

Question 1b

\(\angle C \cong \angle F\)

Question 2a

• A ASA Congruence Theorem
• B AAS Congruence Theorem (Correct answer)
• C SAS Congruence Theorem
• D HL Congruence Theorem

Question 2b

• A \(\angle A \cong \angle B\)
• B \(\angle MCA \cong \angle MCB\)
• C \(\angle C \cong \angle C\)
• D \(\overline{MC} \cong \overline{MC}\) (Correct answer)