Angles in a Triangle Worksheet | KS3/KS4 Maths | Beyond - Free Printable

I'll solve the problems from both worksheets step by step.

## Worksheet A (Left Side)

Problem 1: Find the missing angle marked x in each triangle

a) Triangle with angles 31°, 42°, and x
- Sum of angles in a triangle = 180°
- 31° + 42° + x = 180°
- 73° + x = 180°
- x = 180° - 73°
- x = 107°

b) Triangle with angles 68°, x, and another angle
- Looking at the diagram, this appears to be an isosceles triangle
- If it's isosceles with two equal angles, and one angle is 68°
- The other two angles would be equal: x + x + 68° = 180°
- 2x = 180° - 68°
- 2x = 112°
- x = 56°

c) Triangle with angle 49° and two equal sides (isosceles)
- In an isosceles triangle, the base angles are equal
- Let's call the equal angles x
- x + x + 49° = 180°
- 2x = 180° - 49°
- 2x = 131°
- x = 65.5°

d) Triangle with angle 42° and two equal sides (isosceles)
- The two base angles are equal
- 42° + x + x = 180°
- 2x = 180° - 42°
- 2x = 138°
- x = 69°

Problem 2: Find the size of the angle marked x

Looking at the triangle with exterior angle:
- The exterior angle equals the sum of the two opposite interior angles
- Or we can use: interior angle + exterior angle = 180° (they form a straight line)
- From the diagram, if we have an interior angle and need to find x which appears to be an exterior angle
- x = 180° - interior angle

Without seeing the exact measurements clearly, let me work with what's visible:
If there's a triangle with known angles and we need to find an exterior angle x:
- First find the third interior angle: 180° - (sum of two known angles)
- Then x = 180° - that interior angle

Let me recalculate based on typical problems:
If the triangle has angles like 48° and 24° shown:
- Third angle = 180° - 48° - 24° = 108°
- If x is the exterior angle adjacent to the 108° angle:
- x = 180° - 108° = 72°

Or if x is exterior to a different angle, I need to see which angle it's adjacent to.

Problem 3: Triangles ABD and BCD are isosceles. Is triangle ACD an isosceles triangle?

For triangle ABD (isosceles): Two sides are equal, so two angles are equal
For triangle BCD (isosceles): Two sides are equal, so two angles are equal

To determine if ACD is isosceles, we need to check if any two sides or angles are equal.

Given angle at C is 30°, and using properties of isosceles triangles:
- In △ABD: if AB = BD, then ∠BAD = ∠BDA
- In △BCD: if BC = BD, then ∠BCD = ∠BDC = 30°

Since ∠BCD = 30° and △BCD is isosceles:
- ∠BDC = 30°
- ∠CBD = 180° - 30° - 30° = 120°

Now for △ABD being isosceles, we need more information about which sides are equal.

Actually, looking at the combined figure ACD:
- We need to find if AC = CD or AD = CD or AC = AD
- Or equivalently, if any two angles in △ACD are equal

Angle ACD = 30° (given)
Angle ADC = angle ADB + angle BDC

This requires knowing the specific configuration. Without clear markings showing which sides are equal in each isosceles triangle, I cannot definitively answer without assumptions.

However, typically in such problems, if both ABD and BCD share side BD and are both isosceles with BD as one of the equal sides, then:
- AB = BD and BC = BD, making AB = BC
- This would make △ABC isosceles, but we're asked about △ACD

Let me reconsider: If the problem states both are isosceles and asks about ACD, there's likely a specific relationship.

Given the typical nature of such problems and angle C = 30°, triangle ACD is not necessarily isosceles unless additional conditions are met. The answer depends on which sides are equal in triangles ABD and BCD.

Problem 4: Prove that if you join the opposite corners of a parallelogram, you get congruent triangles

When you draw a diagonal in a parallelogram, it creates two triangles.

Proof:
- In parallelogram ABCD, draw diagonal AC
- Consider triangles ABC and CDA
- AB = CD (opposite sides of parallelogram are equal)
- BC = DA (opposite sides of parallelogram are equal)
- AC = AC (common side)
- Therefore, △ABC ≅ △CDA by SSS (Side-Side-Side) congruence

Alternatively using SAS:
- AB = CD (opposite sides equal)
- BAC = ∠DCA (alternate interior angles, since AB || CD)
- AC = AC (common)
- Therefore △ABC △CDA by SAS

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## Worksheet B (Right Side)

Problem 1: Find the missing angle marked x in each triangle

a) Triangle with angles 31°, 43°, and x
- 31° + 43° + x = 180°
- 74° + x = 180°
- x = 106°

b) Triangle with angles 84°, 57°, and x
- 84° + 57° + x = 180°
- 141° + x = 180°
- x = 39°

c) Isosceles triangle with base angle 37° and vertex angle x
- Since it's isosceles, the two base angles are equal: both 37°
- 37° + 37° + x = 180°
- 74° + x = 180°
- x = 106°

d) Triangle with interior angles 42° and another angle, with exterior angle 110° and unknown exterior angle x
- The interior angle adjacent to 110° exterior angle = 180° - 110° = 70°
- So the triangle has angles: 42°, 70°, and third angle
- Third angle = 180° - 42° - 70° = 68°
- The exterior angle x is adjacent to this 68° angle
- x = 180° - 68° = 112°

Alternatively, using exterior angle theorem:
- Exterior angle = sum of two opposite interior angles
- 110° = 42° + (angle at top)
- Angle at top = 110° - 42° = 68°
- x = 42° + 68° = 110°... wait, let me reconsider

Actually, looking at the diagram more carefully:
- One exterior angle is 110°
- One interior angle is 42°
- x is another exterior angle

Using exterior angle property:
- Interior angle adjacent to 110° = 180° - 110° = 70°
- Third interior angle = 180° - 42° - 70° = 68°
- x (exterior) = 180° - 68° = 112°

Or: x = 42° + 70° = 112° (exterior angle equals sum of two opposite interior angles)

Problem 2: Find the size of the angle marked x

Triangle with angles 48° and 24°, and x appears to be an exterior angle.

- Third interior angle = 180° - 48° - 24° = 108°
- If x is the exterior angle adjacent to the 108° angle:
- x = 180° - 108° = 72°

Or using exterior angle theorem:
- x = 48° + 24° = 72°

Both methods give the same answer.

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Final Answer:

Worksheet A:
1. a) x = 107°
b) x = 56°
c) x = 65.5°
d) x = 69°

2. x = 72° (assuming standard configuration with interior angles 48° and 24°)

3. Cannot be determined as isosceles without knowing which specific sides are equal in triangles ABD and BCD. Additional information needed.

4. Proof: Drawing a diagonal in a parallelogram creates two congruent triangles by SSS or SAS congruence (opposite sides are equal, and they share the diagonal).

Worksheet B:
1. a) x = 106°
b) x = 39°
c) x = 106°
d) x = 112°

2. x = 72°