Geometry Worksheets | Geometry Worksheets - Free Printable

Let’s solve each problem step by step.

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Problem 1: Find t

We have a triangle with angles 65°, 58°, and the third angle is 180° - 65° - 58° = 57°.

Side opposite 57° is t, side opposite 58° is 51.

Use Law of Sines:
t / sin(57°) = 51 / sin(58°)

Calculate:

sin(57°) ≈ 0.8387
sin(58°) ≈ 0.8480

t = 51 × (0.8387 / 0.8480) ≈ 51 × 0.9890 ≈ 50.439 → 50.4

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Problem 2: Find y

Triangle has base split into two parts by height. Left part is 6, angle at bottom left is 46°, top angle is 52°.

First, find the height using left right triangle:

tan(46°) = height / 6 → height = 6 × tan(46°) ≈ 6 × 1.0355 ≈ 6.213

Now, in the right triangle on the right, we know the height is 6.213, and the top angle is 52°, so the bottom right angle is 90° - 52° = 38°? Wait — actually, the 52° is at the top vertex, so in the right triangle on the right, the angle at the top is 52°, so the side adjacent to it is the height, and the hypotenuse is y.

So cos(52°) = adjacent/hypotenuse = height / y

→ y = height / cos(52°) ≈ 6.213 / 0.6157 ≈ 10.09 → 10.1

Wait — let me double-check.

Actually, looking again: The 52° is at the top vertex of the whole triangle, and the dashed line is the height, splitting the triangle into two right triangles.

In the left right triangle: angle at bottom left is 46°, base = 6, so height = 6 * tan(46°) ≈ 6.213 — correct.

In the right right triangle: the top angle is still 52°? No — wait, the 52° is the entire top angle of the big triangle. But when you drop the height, it splits the top angle? Not necessarily — unless it's isosceles, which it’s not.

Actually, re-examining: The 52° is labeled at the top vertex of the big triangle, and the dashed line goes straight down to the base, making two right triangles. So the 52° is split? Or is it entirely on one side?

Looking at the diagram description: In problem 2, the 52° is at the top, and the dashed line is perpendicular to the base, so it creates two right triangles. The 52° is the angle at the top for the whole triangle, but since the height is drawn from that vertex, it splits the 52° into two parts? Actually, no — in standard diagrams like this, if the angle is labeled at the top and the height is drawn from that vertex, then the 52° is the angle between the two sides, and the height splits it only if specified. But here, the 52° is likely the angle at the top vertex of the whole triangle, and the height is drawn from there, so it does split the 52° into two angles — but we don’t know how.

Wait — perhaps I misread. Let me think differently.

In the left right triangle: we have angle 46° at bottom left, base 6, so height h = 6 * tan(46°) ≈ 6.213

Now, in the right right triangle, we need to find y, which is the hypotenuse of that right triangle. We know the height is 6.213, and we need another piece of info.

What is the angle at the top for the right triangle? Since the whole top angle is 52°, and the height is drawn from that vertex, it splits the 52° into two angles: one for the left triangle and one for the right.

In the left triangle, the angles are: 46° at bottom left, 90° at foot of height, so the top angle for the left triangle is 180° - 46° - 90° = 44°.

Therefore, the top angle for the right triangle is 52° - 44° = 8°.

Then, in the right triangle, we have angle at top = 8°, opposite side is the base of the right triangle, adjacent is height = 6.213, hypotenuse is y.

So cos(8°) = adjacent/hypotenuse = 6.213 / y

→ y = 6.213 / cos(8°) ≈ 6.213 / 0.9903 ≈ 6.274 → 6.3

But that seems too small. Let me check the diagram again.

Perhaps the 52° is not split. Maybe the 52° is the angle in the right triangle? Looking back at the user's image description: "2) y = ___" with a triangle, top angle 52°, left bottom angle 46°, base left part 6, dashed height.

I think I made a mistake. In many such problems, the 52° is the angle at the top for the whole triangle, but when you draw the height, it doesn't necessarily split it equally. However, in this case, since we have the left triangle with known angle and side, we can find the height, and then for the right triangle, we need the angle at the top for that triangle.

As above, in left triangle: angles 46°, 90°, so top angle is 44°. Total top angle is 52°, so right triangle top angle is 52° - 44° = 8°.

Then in right triangle: tan(8°) = opposite/adjacent = base_right / height

But we don't need base_right; we need hypotenuse y.

cos(8°) = adjacent/hypotenuse = height / y

y = height / cos(8°) = 6.213 / 0.9903 ≈ 6.274 → 6.3

But let's verify with another approach.

The whole triangle: angles at bottom are 46° and let's call the other β, top is 52°, so 46 + β + 52 = 180 → β = 82°.

Oh! That's better. I forgot that the sum of angles in a triangle is 180°.

So in the big triangle: angles are 46° at left bottom, 52° at top, so right bottom angle is 180 - 46 - 52 = 82°.

Now, the height is drawn from the top vertex to the base, creating two right triangles.

Left right triangle: angles 46°, 90°, so the angle at the top for this triangle is 44° (as before).

Right right triangle: angles at bottom is 82°, at foot is 90°, so angle at top is 180 - 82 - 90 = 8° — same as before.

So yes, y is the hypotenuse of the right triangle with angle 8° at top, adjacent side height = 6.213.

y = height / cos(8°) = 6.213 / cos(8°)

cos(8°) = cos(8) ≈ 0.990268

y ≈ 6.213 / 0.990268 ≈ 6.274 → 6.3

But let's calculate height more accurately.

tan(46°) = tan(46) = let's use calculator value: tan(46) = 1.03553031379

height = 6 * 1.03553031379 = 6.21318188274

cos(8°) = cos(8) = 0.990268068742

y = 6.21318188274 / 0.990268068742 ≈ 6.2742 → rounds to 6.3

Okay, so y = 6.3

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Problem 3: Find u

Triangle with left side 21, bottom left angle 40°, bottom right angle 60°, and u is the base.

First, find the third angle: 180 - 40 - 60 = 80°.

Use Law of Sines: u / sin(80°) = 21 / sin(60°)

sin(80°) ≈ 0.9848, sin(60°) ≈ 0.8660

u = 21 * (0.9848 / 0.8660) ≈ 21 * 1.1372 ≈ 23.881 → 23.9

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Problem 4: Find g

Triangle with left side g, bottom left angle 67°, top angle 46°, right side 40.

First, find the third angle: 180 - 67 - 46 = 67°.

Oh! So angles are 67°, 46°, 67° — so it's isosceles with two 67° angles, so sides opposite are equal.

Side opposite first 67° is the right side, which is 40, so side opposite the other 67° should also be 40, but that's g? Let's see.

Vertices: let's say A at bottom left, B at top, C at bottom right.

Angle at A: 67°, angle at B: 46°, angle at C: 67°.

Side opposite A is BC, opposite B is AC, opposite C is AB.

Given: side AC = 40? The diagram says "40" on the right side, which is probably BC or AB.

Typically, in such diagrams, the side labeled is between the vertices.

Assume: from bottom left to top is one side, top to bottom right is another, bottom left to bottom right is base g.

The 40 is on the side from top to bottom right, so that's side AC if A is bottom left, C is bottom right, B is top.

Standard: let’s define:

- Vertex A: bottom left, angle 67°

- Vertex B: top, angle 46°

- Vertex C: bottom right, angle 67°

Then side opposite A is BC, opposite B is AC, opposite C is AB.

The side labeled 40 is likely BC or AB. Since it's on the right, probably BC, which is opposite A.

Angle at A is 67°, opposite side is BC.

Angle at C is 67°, opposite side is AB.

Since angles at A and C are both 67°, sides opposite them should be equal: so BC = AB.

But the diagram shows 40 on the right side, which is probably BC (from B to C), so BC = 40.

Then AB should also be 40, but we need g, which is the base, AC.

AC is opposite angle B, which is 46°.

So by Law of Sines: AC / sin(46°) = BC / sin(67°)

g / sin(46°) = 40 / sin(67°)

sin(46°) ≈ 0.7193, sin(67°) ≈ 0.9205

g = 40 * (0.7193 / 0.9205) ≈ 40 * 0.7814 ≈ 31.256 → 31.3

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Problem 5: Find Area

Triangle with bottom left angle 69°, top angle 55°, base 36, and height drawn.

First, find the third angle: 180 - 69 - 55 = 56°.

But we have the base = 36, and we need height to find area.

The height is drawn from the top vertex to the base, splitting the base into two parts.

Let me denote: base BC = 36, A is top vertex.

Angle at B is 69°, angle at A is 55°, so angle at C is 56°.

Height from A to BC, let's call foot D.

In triangle ABD, angle at B is 69°, angle at D is 90°, so angle at A for this triangle is 21°.

Similarly, in triangle ADC, angle at C is 56°, angle at D is 90°, so angle at A is 34°.

Total angle at A is 21° + 34° = 55° — good.

Now, let BD = x, DC = 36 - x.

In triangle ABD: tan(69°) = height / x → height = x * tan(69°)

In triangle ADC: tan(56°) = height / (36 - x) → height = (36 - x) * tan(56°)

Set equal: x * tan(69°) = (36 - x) * tan(56°)

tan(69°) ≈ 2.6051, tan(56°) ≈ 1.4826

So: 2.6051x = 1.4826*(36 - x)

2.6051x = 53.3736 - 1.4826x

2.6051x + 1.4826x = 53.3736

4.0877x = 53.3736

x ≈ 53.3736 / 4.0877 ≈ 13.057

Then height = x * tan(69°) ≈ 13.057 * 2.6051 ≈ 34.015

Area = (1/2) * base * height = 0.5 * 36 * 34.015 ≈ 18 * 34.015 ≈ 612.27 → 612.3

Using the other expression: height = (36 - x) * tan(56°) = (36 - 13.057) * 1.4826 ≈ 22.943 * 1.4826 ≈ 34.015 — same.

Area = 0.5 * 36 * 34.015 = 18 * 34.015 = let's compute: 18*34 = 612, 18*0.015=0.27, so 612.27 → 612.3

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Problem 6: Find Area

Triangle with bottom left angle 36°, bottom right angle 58°, base 24, height drawn.

Third angle: 180 - 36 - 58 = 86°.

Base BC = 24, A top vertex.

Height from A to BC, foot D.

In triangle ABD: angle at B is 36°, so angle at A is 54° (since 90-36).

In triangle ADC: angle at C is 58°, so angle at A is 32°.

Total angle at A: 54+32=86° — good.

Let BD = x, DC = 24 - x.

In ABD: tan(36°) = height / x → height = x * tan(36°)

In ADC: tan(58°) = height / (24 - x) → height = (24 - x) * tan(58°)

Set equal: x * tan(36°) = (24 - x) * tan(58°)

tan(36°) ≈ 0.7265, tan(58°) ≈ 1.6003

0.7265x = 1.6003*(24 - x)

0.7265x = 38.4072 - 1.6003x

0.7265x + 1.6003x = 38.4072

2.3268x = 38.4072

x ≈ 38.4072 / 2.3268 ≈ 16.507

height = x * tan(36°) ≈ 16.507 * 0.7265 ≈ 11.992

Area = 0.5 * 24 * 11.992 = 12 * 11.992 ≈ 143.904 → 143.9

Check with other: height = (24 - x) * tan(58°) = (24 - 16.507) * 1.6003 ≈ 7.493 * 1.6003 ≈ 11.992 — same.

Area = 12 * 11.992 = 143.904 → 143.9

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Final Answers:

1) t = 50.4
2) y = 6.3
3) u = 23.9
4) g = 31.3
5) Area = 612.3
6) Area = 143.9

Note: For problem 2, I initially thought it might be different, but after careful calculation, y=6.3 is correct based on the angles.

But let me double-check problem 2 once more.

In problem 2: triangle with bottom left angle 46°, top angle 52°, so bottom right angle is 180-46-52=82°.

Height from top to base, length h.

In left right triangle: h = 6 * tan(46°) ≈ 6 * 1.0355 = 6.213

In right right triangle: the angle at the bottom right is 82°, so in that right triangle, angle at bottom is 82°, so tan(82°) = h / base_right → base_right = h / tan(82°)

But we need y, which is the hypotenuse of the right triangle on the right.

In that triangle, angle at bottom is 82°, adjacent is base_right, opposite is h, hypotenuse is y.

So sin(82°) = opposite/hypotenuse = h / y

Thus y = h / sin(82°)

sin(82°) ≈ 0.9903

y = 6.213 / 0.9903 ≈ 6.274 → still 6.3

cos(82°) = adjacent/hypotenuse = base_right / y, but we don't need that.

Since angle at top for the right triangle is 8°, as before, and cos(8°) = h / y, same thing.

So yes, y=6.3 is correct.

All answers verified.

Final Answer:
1) 50.4
2) 6.3
3) 23.9
4) 31.3
5) 612.3
6) 143.9