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How to solve a triangle given two angles and one side.

 This topic is part of the TCS FREE high school mathematics 'How-to Library'. It shows you how find the unknown angle and sides of a triangle when given two angles and one side length. (See the index page for a list of all available topics in the library.)Â To make best use of this topic, you need to download the Maths Helper Plus software. Click here for instructions.

### Theory:

This triangle has internal angles 'A', 'B' and 'C', and sides of length 'a', 'b' and 'c':

If three of these six measurements are known, then it may be possible to find the other three.

This is called 'solving' the triangle, and this topic will show you how to solve triangles for the remaining angle and sides when two angles and one side are given.

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These are the formulas used to solve triangles:

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1. The sum of the internal angles equals 180º ...

A + B + C = 180º

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2. The 'sine rule' ...

3. The 'cosine rule' ...

Â Â Â Â Â Â  a² = b² + c² - 2bc cosA

Â Â Â  or

Â Â Â Â Â Â  b² = a² + c² - 2ac cosB

Â Â Â  or

Â Â Â Â Â Â  c² = b² + a² - 2ba cosC

We will now use an example to show how these rules are applied to solve a triangle when two angles and one side length are given.

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Example: Solve this triangle for the unknown internal angle and sides:

Step 1: Calculate the unknown angle.Â

The sum of the internal angles equals 180º ...

Â Â Â Â Â Â Â  A + B + C = 180º

so

Â Â Â Â Â Â Â  C = 180º - (A+B)Â

Â Â Â Â Â Â Â Â Â Â  =Â  180º - (30º+70º)

Â Â Â Â Â Â Â Â Â Â  =Â  180º - 100º

Â Â Â Â Â Â Â Â Â Â  =Â  80º

Step 2: Use the sine rule to find one unknown side.Â

NOTE: The sine rule is easier to use than the cosine rule.

To find side 'c' with the sine rule:

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Step 3: Use the sine rule again to find the other unknown side.Â

To find side 'a' with the sine rule:

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The triangle is now solved. This diagram shows all of the sides and angles:

The Method section below shows you how Maths Helper Plus can easily solve your triangles, creating both a labelled diagram and full working steps.

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### Method:

Maths Helper Plus can solve a triangle given two angles and one side length. Full working steps and a labelled diagram are created. The steps below will show you how...

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NOTE: This document has already been set up to solve the example triangle as described in the 'theory' section of this topic.

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#### Step 2Â  Display the triangle solver options box

Double click the mouse in the border to the left of the calculations. ( This area is shaded pale blue in the diagram below.) The triangle solver options box will display its 'Lengths & Angles' tab...

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Click the 'Clear' button to remove the previous triangle, then click on the 'b' edit box. Type the new length for side 'b' of your triangle. Click on the 'A' edit box to type angle 'A', then the 'B' edit box to type angle 'B'.

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NOTE: You can also use other edit boxes to enter the values, eg side 'a', and angles 'A' and 'C'. While the answers will be the same, the letter names will be different and the triangle will be drawn in a different orientation.

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Click the 'Apply' button at the bottom of the edit box. The calculated values will display on the options box.

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Click the 'OK' button to close the options box. The calculations and triangle diagram will be displayed on your screen.

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#### Step 3Â  Adjust the size of the diagram

If the triangle diagram is too big to display properly on your computer screen, briefly press the F10 key to reduce its size. To make the diagram bigger, hold down a Ctrl key while you press F10.

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