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1.4 Key Concepts

1.4 Key Concepts. Angle. Two different Rays with the same Endpoint. Sides. The Rays of the Angle. Vertex. The Endpoint of the Angle. Acute Angle. An Angle between 0° and 90 °. Right Angle. An Angle that equals 90 °. Obtuse Angle. An Angle between 90 ° and 180 °. Straight Angle.

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1.4 Key Concepts

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  1. 1.4 Key Concepts

  2. Angle Two different Rays with the same Endpoint

  3. Sides The Rays of the Angle

  4. Vertex The Endpoint of the Angle

  5. Acute Angle An Angle between 0° and 90 °.

  6. Right Angle An Angle that equals 90 °

  7. Obtuse Angle An Angle between 90 ° and 180 °

  8. Straight Angle An Angle that equals 180 °

  9. Protractor Postulate The measure of an angle is equal to the absolute value of the difference between values of the rays on the protractor.

  10. Angle Addition Postulate If P is in the interior of <RST, then m<RST = m<RSP + m<PST

  11. Name the three angles in the diagram. WXY, or YXW YXZ, or ZXY WXZ, or ZXW You should not name any of these angles X because all three angles have Xas their vertex. EXAMPLE 1 Name angles

  12. Use the diagram to find the measure of the indicated angle. Then classify the angle. a.KHJ b.GHK c.GHJ d.GHL EXAMPLE 2 Measure and classify angles SOLUTION A protractor has an inner and an outer scale. When you measure an angle, check to see which scale to use.

  13. o a. HJ is lined up with the 0 on the inner scale of the protractor. HKpasses through 55 on the inner scale. So, mKHJ = 55 . It is an acute angle. o o o b. HG is lined up with the 0 on the outer scale and HKpasses through 125 on the outer scale. So, mGHK = 125 . It is an obtuse angle. o o o c. m GHJ = 180. It is a straight angle. o d. m GHL= 90. It is a right angle. EXAMPLE 2 Measure and classify angles

  14. 1. Name all the angles in the diagram. Which angle is a right angle? ANSWER PQR , PQS, RQS ; PQSis a right angle . for Examples 1and 2 GUIDED PRACTICE

  15. ANSWER Straight Angle for Examples 1and 2 GUIDED PRACTICE 2. Draw a pair of opposite rays. What type of angle do the rays form?

  16. o ALGEBRAGiven that m LKN =145 , find m LKM andm MKN. STEP 1 Write and solve an equation to find the value of x. mLKN = m LKM + mMKN o o o 145 = (2x + 10)+ (4x – 3) EXAMPLE 3 Find angle measures SOLUTION Angle Addition Postulate Substitute angle measures. 145 = 6x + 7 Combine like terms. 138 = 6x Subtract 7 from each side. 23 = x Divide each side by 6.

  17. STEP 2 Evaluate the given expressions when x = 23. mLKM = (2x+ 10)° = (2 23+ 10)° = 56° mMKN = (4x– 3)° = (4 23– 3)° = 89° So, m LKM = 56°and m MKN = 89°. ANSWER EXAMPLE 3 Find angle measures

  18. Find the indicated angle measures. 3. Given that KLMis a straight angle, find mKLN andm NLM. ANSWER 125°, 55° for Example 3 GUIDED PRACTICE

  19. 4. Given that EFGis a right angle, find mEFH andm HFG. ANSWER 60°, 30° for Example 3 GUIDED PRACTICE

  20. Trapeze The photograph shows some of the angles formed by the ropes in a trapeze apparatus. Identify the congruent angles. If m DEG = 157° ,what is m GKL? SOLUTION There are two pairs of congruent angles: DEF JKL and DEG GKL. Because ∠ DEG GKL, DEG = mGKL. So, mGKL = 157°. ~ ~ ~ EXAMPLE 4 Identify congruent angles

  21. Use the diagram shown. ANSWER T and S, P and R. for Example 4 GUIDED PRACTICE 5. Identify all pairs of congruent angles in the diagram.

  22. Use the diagram shown. o o 6. In the diagram, mPQR = 130 , mQRS = 84, and m TSR = 121 . Find the other angle measures in the diagram. o m PTS = 121°, m QPT = 84° ANSWER for Example 4 GUIDED PRACTICE

  23. In the diagram at the right, YWbisects XYZ, and mXYW = 18. Find m XYZ. o By the Angle Addition Postulate, m XYZ =mXYW + m WYZ. BecauseYW bisects XYZyou know thatXYW WYZ. So, m XYW = m WYZ, and you can write m XYZ = m XYW + m WYZ = 18° + 18° = 36°. ~ EXAMPLE 5 Double an angle measure SOLUTION

  24. ANSWER 90° for Example 5 GUIDED PRACTICE 7. Angle MNPis a straight angle, and NQbisects MNP. Draw MNP And NQ. Use arcs to mark the congruent angles in your diagram, and give the angle measures of these congruent angles.

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