Math Problem Statement

For each triangle find the missing angle, with sides and one angle provided for each triangle.

Solution

The goal here is to find the missing angles in each triangle using the provided side lengths. For triangles where two sides and an angle are known, we'll often use trigonometric laws, like the Law of Cosines or Law of Sines, depending on the given values.

Let's start by solving each triangle step by step:

1. Triangle 1:

Given sides: 8cm,10cm,18cm8 \, \text{cm}, 10 \, \text{cm}, 18 \, \text{cm}
We can use the Cosine Rule to find angle xx.

Cosine rule: cos(x)=a2+b2c22ab\cos(x) = \frac{a^2 + b^2 - c^2}{2ab} where a=8cm,b=10cm,c=18cma = 8 \, \text{cm}, b = 10 \, \text{cm}, c = 18 \, \text{cm}.

Substitute the values to solve for xx.

2. Triangle 2:

Given sides: 9cm,5cm,13cm9 \, \text{cm}, 5 \, \text{cm}, 13 \, \text{cm}
Again, we use the Cosine Rule to find angle γ\gamma.

cos(γ)=a2+b2c22ab\cos(\gamma) = \frac{a^2 + b^2 - c^2}{2ab} where a=9cm,b=5cm,c=13cma = 9 \, \text{cm}, b = 5 \, \text{cm}, c = 13 \, \text{cm}.

3. Triangle 3:

Given sides: 5cm,3cm,7cm5 \, \text{cm}, 3 \, \text{cm}, 7 \, \text{cm}
Use the Cosine Rule to find angle aa.

cos(a)=52+32722×5×3\cos(a) = \frac{5^2 + 3^2 - 7^2}{2 \times 5 \times 3}

4. Triangle 4:

Given sides: 4cm,11cm,9cm4 \, \text{cm}, 11 \, \text{cm}, 9 \, \text{cm}
Find angle bb using the Cosine Rule.

cos(b)=42+921122×4×9\cos(b) = \frac{4^2 + 9^2 - 11^2}{2 \times 4 \times 9}

5. Triangle 5:

Given sides: 7cm,9cm,15cm7 \, \text{cm}, 9 \, \text{cm}, 15 \, \text{cm}
Use the Cosine Rule to find angle

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Triangles
Cosine Rule

Formulas

Cosine Rule: cos(C) = (a^2 + b^2 - c^2) / 2ab
Sum of angles in a triangle: 180°

Theorems

Cosine Rule

Suitable Grade Level

Grades 9-12