Question

triangle congruence: asa and aas
what additional information could be used to prove that the triangles are congruent using aas? choose two correct answers.
ac = 3.9 cm and rq = 3.9 cm
ab = 2.5 cm and mq = 2.5 cm
m∠c = 35° and m∠q = 35°
\overline{cb} \cong \overline{qm}
∠c \cong ∠q

Explanation:

Step1: Recall AAS Congruence

AAS (Angle - Angle - Side) congruence criterion states that if two angles and a non - included side of one triangle are congruent to the corresponding two angles and non - included side of another triangle, then the triangles are congruent.

First, let's find the third angle of each triangle. In $\triangle ABC$, we know that the sum of angles in a triangle is $180^{\circ}$. Given $\angle A = 29^{\circ}$ and $\angle B=116^{\circ}$, so $\angle C=180-(29 + 116)=35^{\circ}$. In $\triangle RMQ$, $\angle R = 29^{\circ}$ and $\angle M = 116^{\circ}$, so $\angle Q=180-(29 + 116)=35^{\circ}$.

Step2: Analyze each option

- Option 1: $AC = 3.9$ cm and $RQ=3.9$ cm. $AC$ is a side in $\triangle ABC$ and $RQ$ is a side in $\triangle RMQ$. But for AAS, we need two angles and a non - included side. Here, we have a side and we need to check the angles. Wait, $\angle A=\angle R = 29^{\circ}$, $\angle C=\angle Q = 35^{\circ}$, and if $AC = RQ$, but $AC$ is opposite $\angle B$ and $RQ$ is opposite $\angle M$. Wait, maybe I made a mistake. Wait, let's re - evaluate.
- Option 2: $AB = 2.5$ cm and $MQ = 2.5$ cm. $AB$ is a side in $\triangle ABC$ (between $\angle A$ and $\angle B$) and $MQ$ is a side in $\triangle RMQ$ (between $\angle M$ and $\angle Q$). Not a non - included side for AAS.
- Option 3: $m\angle C = 35^{\circ}$ and $m\angle Q=35^{\circ}$. We already know these angles are equal from the angle sum property. This is just confirming the angles, not giving a side for AAS. Wait, no, actually, if we know two angles (we already know $\angle A=\angle R = 29^{\circ}$, and now $\angle C=\angle Q = 35^{\circ}$) and if we have a side, but this option just gives the angles. Wait, maybe the first correct option is the one with $m\angle C = 35^{\circ}$ and $m\angle Q = 35^{\circ}$ (since we already have $\angle A=\angle R$) and then another option with a side. Wait, the first triangle has angles $29^{\circ},116^{\circ},35^{\circ}$ and the second has $29^{\circ},116^{\circ},35^{\circ}$. So for AAS, we can have:
- Option with $m\angle C = 35^{\circ}$ and $m\angle Q = 35^{\circ}$ (confirming the third angle) and another option like $AB = MQ$ (but $AB$ is between $\angle A$ and $\angle B$, $MQ$ is between $\angle M$ and $\angle Q$). Wait, no. Wait, the correct options should be the one where we have two angles (which we already know are equal) and a non - included side.
- The option with $m\angle C = 35^{\circ}$ and $m\angle Q = 35^{\circ}$ (since we already have $\angle A=\angle R$) and the option with $AC = RQ$ (but $AC$ is opposite $\angle B$ and $RQ$ is opposite $\angle M$). Wait, maybe the two correct options are:
- The option with $m\angle C = 35^{\circ}$ and $m\angle Q = 35^{\circ}$ (because we already have $\angle A=\angle R$) and the option with $AB = MQ$? No, wait. Wait, let's start over.

We know that $\angle A=\angle R = 29^{\circ}$, $\angle C=\angle Q = 35^{\circ}$. For AAS, we need a side that is not included between the two angles. In $\triangle ABC$, the sides: $AB$ is between $\angle A$ and $\angle B$, $BC$ is between $\angle B$ and $\angle C$, $AC$ is between $\angle A$ and $\angle C$. In $\triangle RMQ$, $RM$ is between $\angle R$ and $\angle M$, $MQ$ is between $\angle M$ and $\angle Q$, $RQ$ is between $\angle R$ and $\angle Q$.

So, if we take $\angle A=\angle R$, $\angle C=\angle Q$, and $AB = MQ$ ( $AB$ is opposite $\angle C$ and $MQ$ is opposite $\angle R$), or $\angle A=\angle R$, $\angle C=\angle Q$, and $BC = RM$ ( $BC$ is opposite $\angle A$ and $RM$ is opposite $\angle Q$), or $\angle A=\angle R$, $\angle C=\angle Q$, and $AC = RQ$ ( $AC$ is opposite $\angle B$ and $RQ$ is opposite $\angle M$). Wait, the option with $m\angle C = 35^{\circ}$ and $m\angle Q = 35^{\circ}$ (since we already have $\angle A=\angle R$) and the option with $AC = RQ = 3.9$ cm (because $\angle A=\angle R$, $\angle C=\angle Q$, and $AC$ (opposite $\angle B$) and $RQ$ (opposite $\angle M$) are equal) and the option with $AB = MQ=2.5$ cm ( $\angle A=\angle R$, $\angle B=\angle M$, and $AB$ (opposite $\angle C$) and $MQ$ (opposite $\angle Q$) are equal) and the option with $\angle C\cong\angle Q$ (which is the same as $m\angle C = m\angle Q$).

Wait, the two correct options are:
1. $m\angle C = 35^{\circ}$ and $m\angle Q = 35^{\circ}$ (because we already have $\angle A=\angle R$, so with $\angle C=\angle Q$, we have two angles, and then we need a side)
2. $AC = 3.9$ cm and $RQ = 3.9$ cm (since $\angle A=\angle R$, $\angle C=\angle Q$, and $AC$ (a non - included side with respect to $\angle A$ and $\angle C$? Wait, no, $AC$ is between $\angle A$ and $\angle C$? Wait, $\angle A$ and $\angle C$: the side between them is $BC$? No, in $\triangle ABC$, the sides: $AB$ is between $\angle A$ and $\angle B$, $BC$ is between $\angle B$ and $\angle C$, $AC$ is between $\angle A$ and $\angle C$. So $\angle A$ and $\angle C$ include side $AC$. So AAS requires two angles and a non - included side. So if we have $\angle A=\angle R$ (angle), $\angle B=\angle M$ (angle), and $AC = RQ$ (side, non - included between $\angle A$ and $\angle B$? Wait, $\angle A$ and $\angle B$ include side $AB$. So $AC$ is opposite $\angle B$, and $RQ$ is opposite $\angle M$. So since $\angle B=\angle M$, $\angle A=\angle R$, and $AC = RQ$ (opposite angles), this is AAS. Also, the option with $m\angle C = 35^{\circ}$ and $m\angle Q = 35^{\circ}$: since we know $\angle A=\angle R$, and now $\angle C=\angle Q$, so we have two angles, and then a side.

So the two correct options are:
- The option with $m\angle C = 35^{\circ}$ and $m\angle Q = 35^{\circ}$
- The option with $AC = 3.9$ cm and $RQ = 3.9$ cm (or the option with $AB = 2.5$ cm and $MQ = 2.5$ cm? Wait, no, let's check the angle - side relationship.

Wait, in $\triangle ABC$: angles are $\angle A = 29^{\circ}$, $\angle B=116^{\circ}$, $\angle C = 35^{\circ}$. Sides: $AB$ (opposite $\angle C$), $BC$ (opposite $\angle A$), $AC$ (opposite $\angle B$).

In $\triangle RMQ$: angles are $\angle R = 29^{\circ}$, $\angle M=116^{\circ}$, $\angle Q = 35^{\circ}$. Sides: $RM$ (opposite $\angle Q$), $MQ$ (opposite $\angle R$), $RQ$ (opposite $\angle M$).

So for AAS:
- If we take $\angle A=\angle R$ ( $29^{\circ}$), $\angle C=\angle Q$ ( $35^{\circ}$), and $BC = RM$ (since $BC$ is opposite $\angle A$ and $RM$ is opposite $\angle Q$)
- Or $\angle A=\angle R$ ( $29^{\circ}$), $\angle B=\angle M$ ( $116^{\circ}$), and $AC = RQ$ (since $AC$ is opposite $\angle B$ and $RQ$ is opposite $\angle M$)
- Or $\angle B=\angle M$ ( $116^{\circ}$), $\angle C=\angle Q$ ( $35^{\circ}$), and $AB = RM$ (since $AB$ is opposite $\angle C$ and $RM$ is opposite $\angle Q$)

So the option with $m\angle C = 35^{\circ}$ and $m\angle Q = 35^{\circ}$ (gives us the second pair of angles) and the option with $AC = 3.9$ cm and $RQ = 3.9$ cm (gives us the non - included side for the angles $\angle A$ and $\angle B$ (or $\angle R$ and $\angle M$)) and the option with $AB = 2.5$ cm and $MQ = 2.5$ cm (gives us the non - included side for the angles $\angle B$ and $\angle C$ (or $\angle M$ and $\angle Q$)).

But the problem says "Choose two correct answers". Let's analyze each option:

1. $AC = 3.9$ cm and $RQ = 3.9$ cm: $\angle A=\angle R = 29^{\circ}$, $\angle B=\angle M = 116^{\circ}$, and $AC$ (opposite $\angle B$) $= RQ$ (opposite $\angle M$). So by AAS (two angles and a non - included side: $\angle A=\angle R$, $\angle B=\angle M$, and $AC = RQ$ (non - included between $\angle A$ and $\angle B$? Wait, $\angle A$ and $\angle B$ include $AB$, so $AC$ is non - included between $\angle A$ and $\angle B$? No, $\angle A$ and $\angle B$ are adjacent to $AB$, $AC$ is adjacent to $\angle A$ and $\angle C$. Maybe this is a case of AAS where we have $\angle A=\angle R$, $\angle B=\angle M$, and $AC = RQ$ (side opposite $\angle B$ and $\angle M$). So this is valid.

2. $AB = 2.5$ cm and $MQ = 2.5$ cm: $\angle B=\angle M = 116^{\circ}$, $\angle C=\angle Q = 35^{\circ}$, and $AB$ (opposite $\angle C$) $= MQ$ (opposite $\angle R$). Wait, no, $\angle C=\angle Q$, $\angle B=\angle M$, and $AB$ is opposite $\angle C$, $MQ$ is opposite $\angle R$. Not a match. Wait, maybe I messed up.

3. $m\angle C = 35^{\circ}$ and $m\angle Q = 35^{\circ}$: We already calculated that $\angle C=\angle Q = 35^{\circ}$, and we know $\angle A=\angle R = 29^{\circ}$. So with two angles equal, if we have a side, but this option just gives the angles. But when combined with a side option, it works.

4. $\overline{CB}\cong\overline{QM}$: $CB$ is opposite $\angle A$, $QM$ is opposite $\angle R$. $\angle A=\angle R$, so if $CB = QM$, then by AAS ( $\angle A=\angle R$, $\angle B=\angle M$, $CB = QM$).

5. $\angle C\cong\angle Q$: This is the same as $m\angle C = m\angle Q$, which we already know from the angle sum.

So the two correct options are:

- The option with $m\angle C = 35^{\circ}$ and $m\angle Q = 35^{\circ}$ (because it gives the second pair of equal angles)
- The option with $AC = 3.9$ cm and $RQ = 3.9$ cm (because it gives the equal non - included side for the two equal angles $\angle A$ and $\angle B$ (or $\angle R$ and $\angle M$))

Or the option with $AB = 2.5$ cm and $MQ = 2.5$ cm and the angle option. But let's re - check.

Wait, the first triangle: angles $29,116,35$; second triangle: angles $29,116,35$. So we have two pairs of equal angles. For AAS, we need one pair of equal sides that is not included between the two angles.

So if we take the angle option ($m\angle C = 35^{\circ}$ and $m\angle Q = 35^{\circ}$) which confirms the second angle pair, and then a side option like $AC = RQ$ (since $\angle A=\angle R$, $\angle B=\angle M$, and $AC$ is opposite $\angle B$, $RQ$ is opposite $\angle M$) or $AB = MQ$ (since $\angle B=\angle M$, $\angle C=\angle Q$, and $AB$ is opposite $\angle C$, $MQ$ is opposite $\angle R$).

So the two correct options are:

1. $m\angle C = 35^{\circ}$ and $m\angle Q = 35^{\circ}$

2. $AC = 3.9$ cm and $RQ = 3.9$ cm (or $AB = 2.5$ cm and $MQ = 2.5$ cm)

But from the given options, the two correct ones are:

- The option with $m\angle C = 35^{\circ}$ and $m\angle Q = 35^{\circ}$

- The option with $AC = 3.9$ cm and $RQ = 3.9$ cm (or the option with $AB = 2.5$ cm and $MQ = 2.5$ cm)

Answer:

1. $m\angle C = 35^{\circ}$ and $m\angle Q = 35^{\circ}$
2. $AC = 3.9$ cm and $RQ = 3.9$ cm (or $AB = 2.5$ cm and $MQ = 2.5$ cm, depending on the side - angle relationship)