classify each set of conditions by the possible number of triangles, if any, that can be formed with their conditions. drag each set of conditions to the table.
more than one triangle can be formed
exactly one triangle can be formed
no triangles can be formed
(some condition sets with angle and side info, e.g., m∠a = 35°, m∠b = 35°, ab = 3; m∠a = 90°, m∠b = 30°, m∠c = 60°; ab = 5, bc = 6, ca = 3; m∠a = 30°, m∠b = 30°, bc = 5; m∠c = 70°, ca = 5, ab = 7)

Question

Accepted Answer

# Explanation:
## Step1: Analyze first condition set
Given: $m\angle A=45^\circ$, $m\angle B=45^\circ$, $AB=3$
1.  Calculate third angle: $m\angle C = 180^\circ - 45^\circ - 45^\circ = 90^\circ$
2.  With two angles and included side (ASA), only one unique triangle can be formed.

## Step2: Analyze second condition set
Given: $m\angle A=90^\circ$, $m\angle B=30^\circ$, $m\angle C=60^\circ$
1.  Only angle measures are given, no side lengths.
2.  Infinite similar triangles of different sizes can be formed, so more than one triangle exists.

## Step3: Analyze third condition set
Given: $AB=5$, $BC=8$, $CA=3$
1.  Check triangle inequality: $AB + CA = 5 + 3 = 8$, which equals $BC$.
2.  A valid triangle requires the sum of any two sides to be **greater than** the third, so no triangle can be formed.

## Step4: Analyze fourth condition set
Given: $m\angle A=50^\circ$, $m\angle B=50^\circ$, $BC=3$
1.  Calculate third angle: $m\angle C = 180^\circ - 50^\circ - 50^\circ = 80^\circ$
2.  With two angles and a non-included side (AAS), only one unique triangle can be formed.

## Step5: Analyze fifth condition set
Given: $m\angle C=70^\circ$, $CA=5$, $AB=7$
1.  This is the SSA (side-side-angle) case: we have $\angle C$, side $CA$ (adjacent to $\angle C$), and side $AB$ (opposite $\angle C$).
2.  Calculate the height from $B$ to $CA$: $h = AB \cdot \sin(70^\circ) \approx 7 \cdot 0.9397 \approx 6.578$.
3.  Since $CA=5 < h$, no triangle can be formed.

# Answer:
| Category | Conditions |
|----------|------------|
| More than one triangle can be formed | $m\angle A=90^\circ$, $m\angle B=30^\circ$, $m\angle C=60^\circ$ |
| Exactly one triangle can be formed | $m\angle A=45^\circ$, $m\angle B=45^\circ$, $AB=3$; $m\angle A=50^\circ$, $m\angle B=50^\circ$, $BC=3$ |
| No triangles can be formed | $AB=5$, $BC=8$, $CA=3$; $m\angle C=70^\circ$, $CA=5$, $AB=7$ |

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Step1: Analyze first condition set

Step2: Analyze second condition set

Step3: Analyze third condition set

Step4: Analyze fourth condition set

Step5: Analyze fifth condition set

Answer:

Category	Conditions
Exactly one triangle can be formed	$m\angle A=45^\circ$, $m\angle B=45^\circ$, $AB=3$; $m\angle A=50^\circ$, $m\angle B=50^\circ$, $BC=3$
No triangles can be formed	$AB=5$, $BC=8$, $CA=3$; $m\angle C=70^\circ$, $CA=5$, $AB=7$