Isos-and-EquilTriangles2.pdf - Name Hour 4-6 Isosceles and ... - Free Printable

Let’s solve this step by step.

We are given two triangles that are similar — meaning their corresponding angles are equal, and their sides are in proportion.

The first triangle has sides:
- 20 (bottom)
- 16 (left side)
- ? (right side — we’ll call it x for now)

The second triangle has sides:
- ? (bottom — we’ll call it y)
- 12 (left side)
- 15 (right side)

Since the triangles are similar, the ratios of corresponding sides must be equal.

Look at the left sides:
First triangle left = 16
Second triangle left = 12
So the ratio is 16 : 12 → simplify to 4 : 3

That means every side in the first triangle is 4/3 times the corresponding side in the second triangle.

Now check the right sides:
First triangle right = x
Second triangle right = 15
So:
x / 15 = 4 / 3
→ x = (4 × 15) / 3 = 60 / 3 = 20

Wait — but looking back at the diagram, the first triangle’s right side is labeled as “?” and the bottom is 20. The second triangle’s right side is 15, bottom is “?”, left is 12.

Actually, let’s match corresponding sides properly.

In similar triangles, the order matters. Let’s assume:

Triangle ABC ~ Triangle DEF

From the diagram layout:

Top vertex corresponds to top vertex.

Left side of first triangle (16) corresponds to left side of second triangle (12).

Right side of first triangle (?) corresponds to right side of second triangle (15).

Bottom of first triangle (20) corresponds to bottom of second triangle (?).

So:

Ratio = 16 / 12 = 4 / 3

Therefore:

Right side of first triangle = (4/3) × 15 = 20 → so the missing side on the first triangle is 20? But wait — the bottom is already 20. That would mean two sides are 20? Possible, but let’s check consistency.

Alternatively, maybe the correspondence is different.

Wait — look again.

In the first triangle: sides are 20 (base), 16 (left), ? (right)

In the second triangle: sides are ? (base), 12 (left), 15 (right)

If left sides correspond: 16 ↔ 12 → ratio 4:3

Then right sides: ? ↔ 15 → so ? = (4/3)*15 = 20

Then base: 20 ↔ ? → so ? = (3/4)*20 = 15

So second triangle base is 15.

But then the first triangle has sides 20, 16, 20 — which is fine (isosceles?).

Second triangle: 15, 12, 15 — also isosceles? Wait, 12 and 15 don’t match.

Hold on — if left side 16 corresponds to 12, and right side ? corresponds to 15, and base 20 corresponds to ?, then:

Set up proportions:

16 / 12 = ? / 15 = 20 / ?

Let’s use 16/12 = ?/15

Cross multiply: 16 * 15 = 12 * ?

240 = 12 * ?

? = 240 / 12 = 20 → so first triangle’s right side is 20.

Now, 16/12 = 20 / ? (for the bases)

So 4/3 = 20 / ?

Then ? = 20 * 3 / 4 = 60 / 4 = 15

So second triangle’s base is 15.

Thus, the missing values are:

First triangle, right side: 20

Second triangle, base: 15

But wait — in the original problem, the first triangle’s right side is marked with a question mark, and the second triangle’s base is marked with a question mark.

Also, there’s another part: angle measures.

It says: ∠A = 70°, ∠B = 50°, find ∠C.

In any triangle, angles add to 180°.

So ∠C = 180 - 70 - 50 = 60°

And since triangles are similar, corresponding angles are equal.

So in the other triangle, the angles should also be 70°, 50°, 60° — just matched to corresponding vertices.

But the question seems to ask for side lengths primarily.

Looking back at the user’s image description (even though I can’t see it, from context), the main task is to find the missing side lengths using similarity.

So final answers:

Missing side in first triangle (right side): 20

Missing side in second triangle (base): 15

Also, angle C: 60°

But let me double-check the proportion.

Alternative approach: set up ratio between the two triangles.

Scale factor from second to first: since 12 → 16, scale factor = 16/12 = 4/3

So all sides of first triangle = (4/3) × corresponding sides of second.

So:

Right side of first = (4/3) × 15 = 20 ✔️

Base of first = 20 = (4/3) × base of second → so base of second = 20 × 3/4 = 15 ✔️

Perfect.

Angles: sum to 180, so 180 - 70 - 50 = 60° ✔️

Final Answer:
The missing side in the first triangle is 20, the missing side in the second triangle is 15, and angle C is 60 degrees.