9.3 Use properties of angles, triangles, and the pythagorean theorem  (Page 5/15)

Use the Pythagorean Theorem to find the length of the leg.

8

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Use the Pythagorean Theorem to find the length of the leg.

12

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Kelvin is building a gazebo and wants to brace each corner by placing a $\text{10-inch}$ wooden bracket diagonally as shown. How far below the corner should he fasten the bracket if he wants the distances from the corner to each end of the bracket to be equal? Approximate to the nearest tenth of an inch.

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.	the distance from the corner that the bracket should be attached
Step 3. Name. Choose a variable to represent it.	Let x = the distance from the corner
Step 4. Translate. Write the appropriate formula. Substitute.
Step 5. Solve the equation. Isolate the variable. Use the definition of the square root. Simplify. Approximate to the nearest tenth.
Step 6. Check: Yes.
Step 7. Answer the question.	Kelvin should fasten each piece of wood approximately 7.1" from the corner.

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John puts the base of a $\text{13-ft}$ ladder $5$ feet from the wall of his house. How far up the wall does the ladder reach?

12 feet

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Randy wants to attach a $\text{17-ft}$ string of lights to the top of the $\text{15-ft}$ mast of his sailboat. How far from the base of the mast should he attach the end of the light string?

8 feet

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Key concepts

• Supplementary and Complementary Angles
  • If the sum of the measures of two angles is 180°, then the angles are supplementary.
  • If $\text{∠}A$ and $\text{∠}B$ are supplementary, then $m\text{∠}A+m\text{∠}B=180$ .
  • If the sum of the measures of two angles is 90°, then the angles are complementary.
  • If $\text{∠}A$ and $\text{∠}B$ are complementary, then $m\text{∠}A+m\text{∠}B=90$ .
• Solve Geometry Applications
  1. Read the problem and make sure you understand all the words and ideas. Draw a figure and label it with the given information.
  2. Identify what you are looking for.
  3. Name what you are looking for and choose a variable to represent it.
  4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
  5. Solve the equation using good algebra techniques.
  6. Check the answer in the problem and make sure it makes sense.
  7. Answer the question with a complete sentence.
• Sum of the Measures of the Angles of a Triangle

  

  • For any $\text{Δ}ABC,$ the sum of the measures is 180°
  • $m\text{∠}A+m\text{∠}B=180$
• Right Triangle

  

  • A right triangle is a triangle that has one 90° angle, which is often marked with a $\text{⦜}$ symbol.
• Properties of Similar Triangles
  • If two triangles are similar, then their corresponding angle measures are equal and their corresponding side lengths have the same ratio.

Practice makes perfect

Use the Properties of Angles

In the following exercises, find ⓐ the supplement and ⓑ the complement of the given angle.

In the following exercises, use the properties of angles to solve.