1.4 Key Concepts. Angle Two different Rays with the same Endpoint.

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Presentation transcript:

1.4 Key Concepts

Angle Two different Rays with the same Endpoint

Sides T TT The Rays of the Angle

Vertex The Endpoint of the Angle

Acute Angle A AA An Angle between 0° and 90 °.

Right Angle An Angle that equals 90 °

Obtuse Angle An Angle between 90 ° and 180 °

Straight Angle A AA An Angle that equals 180 °

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

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

EXAMPLE 1 Name angles 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 X as their vertex.

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

EXAMPLE 2 Measure and classify angles a. HJ is lined up with the 0 on the inner scale of the protractor. HK passes through 55 on the inner scale. So, m KHJ = 55. It is an acute angle. o o o b. HG is lined up with the 0 on the outer scale and HK passes through 125 on the outer scale. So, m GHK = 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. o

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

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

EXAMPLE 3 Find angle measures o ALGEBRA Given that m LKN =145, find m LKM and m MKN. SOLUTION STEP 1 Write and solve an equation to find the value of x. m LKN = m LKM + m MKN Angle Addition Postulate Substitute angle measures. 145 = 6x + 7 Combine like terms. Subtract 7 from each side. 138 = 6x Divide each side by 6.23 = x 145 = (2x + 10) + (4x – 3) o o o

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

GUIDED PRACTICE for Example 3 Find the indicated angle measures. 3. Given that KLM is a straight angle, find m KLN and m NLM. ANSWER 125°, 55°

GUIDED PRACTICE for Example 3 4. Given that EFG is a right angle, find m EFH and m HFG. ANSWER 60°, 30°

EXAMPLE 4 Identify congruent angles 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? Trapeze SOLUTION There are two pairs of congruent angles: DEF JKL and DEG GKL. ~~ Because  DEG GKL, DEG = m GKL. So, m GKL = 157°. ~

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

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

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

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