Geometry: The Angle Addition Postulate

The Angle Addition Postulate You will learn to find the measure of an angle and the bisector of an angle. NOTHING NEW! The Angle Addition Postulate 1) Draw an acute, an obtuse, or a right angle. Label the angle RST. R S T The Angle Addition Postulate 1) Draw an acute, an obtuse, or a right angle. Label the angle RST. 2) Draw and label a point X in the interior of the angle. Then draw SX. S R X T The Angle Addition Postulate 1) Draw an acute, an obtuse, or a right angle. Label the angle RST. 2) Draw and label a point X in the interior of the angle. Then draw SX. S R X T 3) For each angle, find mRSX, mXST, and RST. The Angle Addition Postulate 1) Draw an acute, an obtuse, or a right angle. Label the angle RST. 2) Draw and label a point X in the interior of the angle. Then draw SX. S R 45° X 30° T 3) For each angle, find mRSX, mXST, and RST. The Angle Addition Postulate 1) Draw an acute, an obtuse, or a right angle. Label the angle RST. 2) Draw and label a point X in the interior of the angle. Then draw SX. S R 45° X 75° 30° T 3) For each angle, find mRSX, mXST, and RST. The Angle Addition Postulate 1) How does the sum of mRSX and mXST compare to mRST ? R 45° X 75° 30° S T The Angle Addition Postulate 1) How does the sum of mRSX and mXST compare to mRST ? Their sum is equal to the measure of RST . mXST = 30 + mRSX = 45 = mRST = 75 R 45° X 75° 30° S T The Angle Addition Postulate 1) How does the sum of mRSX and mXST compare to mRST ? Their sum is equal to the measure of RST . mXST = 30 + mRSX = 45 = mRST = 75 R 2) Make a conjecture about the relationship between the two smaller angles and the larger angle. 45° X 75° 30° S T The Angle Addition Postulate 1) How does the sum of mRSX and mXST compare to mRST ? Their sum is equal to the measure of RST . mXST = 30 + mRSX = 45 = mRST = 75 R 2) Make a conjecture about the relationship between the two smaller angles and the larger angle. The sum of the measures of the two smaller angles is equal to the measure of the larger angle. S 45° X 75° 30° T The Angle Addition Postulate For any angle PQR, if A is in the interior of PQR, then mPQA + mAQR = mPQR. Postulate 3-3 Angle Addition Postulate P 1 Q 2 A R The Angle Addition Postulate For any angle PQR, if A is in the interior of PQR, then mPQA + mAQR = mPQR. Postulate 3-3 Angle Addition Postulate P 1 Q 2 A m1 + m2 = mPQR. R The Angle Addition Postulate For any angle PQR, if A is in the interior of PQR, then mPQA + mAQR = mPQR. Postulate 3-3 Angle Addition Postulate P 1 Q 2 A m1 + m2 = mPQR. R There are two equations that can be derived using Postulate 3 – 3. The Angle Addition Postulate For any angle PQR, if A is in the interior of PQR, then mPQA + mAQR = mPQR. Postulate 3-3 Angle Addition Postulate P 1 Q 2 A m1 + m2 = mPQR. R There are two equations that can be derived using Postulate 3 – 3. m1 = mPQR – m2 m2 = mPQR – m1 The Angle Addition Postulate For any angle PQR, if A is in the interior of PQR, then mPQA + mAQR = mPQR. Postulate 3-3 Angle Addition Postulate P 1 Q 2 A m1 + m2 = mPQR. R There are two equations that can be derived using Postulate 3 – 3. m1 = mPQR – m2 m2 = mPQR – m1 These equations are true no matter where A is located in the interior of PQR. The Angle Addition Postulate Find m2 if mXYZ = 86 and m1 = 22. X 1 Y W 2 Z The Angle Addition Postulate Find m2 if mXYZ = 86 and m1 = 22. m2 + m1 = mXYZ Postulate 3 – 3. X 1 Y W 2 Z The Angle Addition Postulate Find m2 if mXYZ = 86 and m1 = 22. m2 + m1 = mXYZ Postulate 3 – 3. X 1 m2 = mXYZ – m1 Y W 2 Z The Angle Addition Postulate Find m2 if mXYZ = 86 and m1 = 22. m2 + m1 = mXYZ Postulate 3 – 3. X 1 m2 = mXYZ – m1 m2 = 86 – 22 Y W 2 Z The Angle Addition Postulate Find m2 if mXYZ = 86 and m1 = 22. m2 + m1 = mXYZ Postulate 3 – 3. X 1 m2 = mXYZ – m1 m2 = 86 – 22 Y W 2 m2 = 64 Z The Angle Addition Postulate Find mABC and mCBD if mABD = 120. C (5x – 6)° 2x° A B D The Angle Addition Postulate Find mABC and mCBD if mABD = 120. mABC + mCBD = mABD Postulate 3 – 3. C (5x – 6)° 2x° A B D The Angle Addition Postulate Find mABC and mCBD if mABD = 120. mABC + mCBD = mABD 2x + (5x – 6) = 120 Postulate 3 – 3. Substitution C (5x – 6)° 2x° A B D The Angle Addition Postulate Find mABC and mCBD if mABD = 120. mABC + mCBD = mABD 2x + (5x – 6) = 120 Postulate 3 – 3. Substitution 7x – 6 = 120 Combine like terms C (5x – 6)° 2x° A B D The Angle Addition Postulate Find mABC and mCBD if mABD = 120. mABC + mCBD = mABD 2x + (5x – 6) = 120 Postulate 3 – 3. Substitution 7x – 6 = 120 7x = 126 Combine like terms Add 6 to both sides C (5x – 6)° 2x° A B D The Angle Addition Postulate Find mABC and mCBD if mABD = 120. mABC + mCBD = mABD 2x + (5x – 6) = 120 Postulate 3 – 3. Substitution 7x – 6 = 120 7x = 126 x = 18 Combine like terms Add 6 to both sides Divide each side by 7 C (5x – 6)° 2x° A B D The Angle Addition Postulate Find mABC and mCBD if mABD = 120. mABC + mCBD = mABD 2x + (5x – 6) = 120 Postulate 3 – 3. Substitution 7x – 6 = 120 7x = 126 x = 18 Combine like terms Add 6 to both sides Divide each side by 7 C mABC = 2x mABC = 2(18) mABC = 36 A 2x° B (5x – 6)° D The Angle Addition Postulate Find mABC and mCBD if mABD = 120. mABC + mCBD = mABD 2x + (5x – 6) = 120 Postulate 3 – 3. Substitution 7x – 6 = 120 7x = 126 x = 18 Combine like terms Add 6 to both sides Divide each side by 7 C mABC = 2x mABC = 2(18) mABC = 36 mCBD = 5x – 6 mCBD = 5(18) – 6 mCBD = 90 – 6 mCBD = 84 A 2x° B (5x – 6)° D The Angle Addition Postulate Find mABC and mCBD if mABD = 120. mABC + mCBD = mABD 2x + (5x – 6) = 120 Postulate 3 – 3. Substitution 7x – 6 = 120 7x = 126 x = 18 36 + 84 = 120 Combine like terms Add 6 to both sides Divide each side by 7 C mABC = 2x mABC = 2(18) mABC = 36 mCBD = 5x – 6 mCBD = 5(18) – 6 mCBD = 90 – 6 mCBD = 84 A 2x° B (5x – 6)° D The Angle Addition Postulate Just as every segment has a midpoint that bisects the segment, every angle has a ___ that bisects the angle. The Angle Addition Postulate Just as every segment has a midpoint that bisects the segment, every angle ray has a ___ that bisects the angle. The Angle Addition Postulate Just as every segment has a midpoint that bisects the segment, every angle ray has a ___ that bisects the angle. This ray is called an ____________ . The Angle Addition Postulate Just as every segment has a midpoint that bisects the segment, every angle ray has a ___ that bisects the angle. angle bisector This ray is called an ____________ . The Angle Addition Postulate Just as every segment has a midpoint that bisects the segment, every angle ray has a ___ that bisects the angle. angle bisector This ray is called an ____________ . The Angle Addition Postulate The bisector of an angle is the ray with its endpoint at the vertex of the angle, extending into the interior of the angle. The bisector separates the angle into two angles of equal measure. Definition of an Angle Bisector 1 P Q 2 A R The Angle Addition Postulate The bisector of an angle is the ray with its endpoint at the vertex of the angle, extending into the interior of the angle. The bisector separates the angle into two angles of equal measure. Definition of an Angle Bisector 1 P QA is the bisector of PQR. A Q 2 R The Angle Addition Postulate The bisector of an angle is the ray with its endpoint at the vertex of the angle, extending into the interior of the angle. The bisector separates the angle into two angles of equal measure. Definition of an Angle Bisector 1 P QA is the bisector of PQR. A Q 2 m1 = m2 R The Angle Addition Postulate If AT bisects CAN and mCAN = 130, find 1 and 2. N T 2 1 C A The Angle Addition Postulate If AT bisects CAN and mCAN = 130, find 1 and 2. Since AT bisects CAN, 1 = 2. N T 2 1 C A The Angle Addition Postulate If AT bisects CAN and mCAN = 130, find 1 and 2. Since AT bisects CAN, 1 = 2. N 1 + 2 = CAN Postulate 3 - 3 T 2 1 C A The Angle Addition Postulate If AT bisects CAN and mCAN = 130, find 1 and 2. Since AT bisects CAN, 1 = 2. N 1 + 2 = CAN 1 + 2 = 130 Postulate 3 - 3 Replace CAN with 130 T 2 1 C A The Angle Addition Postulate If AT bisects CAN and mCAN = 130, find 1 and 2. Since AT bisects CAN, 1 = 2. N 1 + 2 = CAN 1 + 2 = 130 1 + 1 = 130 Postulate 3 - 3 Replace CAN with 130 Replace 2 with 1 2 T 1 C A The Angle Addition Postulate If AT bisects CAN and mCAN = 130, find 1 and 2. Since AT bisects CAN, 1 = 2. N 1 + 2 = CAN 1 + 2 = 130 1 + 1 = 130 2(1) = 130 Postulate 3 - 3 Replace CAN with 130 Replace 2 with 1 Combine like terms 1 2 T C A The Angle Addition Postulate If AT bisects CAN and mCAN = 130, find 1 and 2. Since AT bisects CAN, 1 = 2. N 1 + 2 = CAN 1 + 2 = 130 1 + 1 = 130 2(1) = 130 (1) = 65 Postulate 3 - 3 Replace CAN with 130 Replace 2 with 1 Combine like terms Divide each side by 2 1 2 T C A The Angle Addition Postulate If AT bisects CAN and mCAN = 130, find 1 and 2. Since AT bisects CAN, 1 = 2. N 1 + 2 = CAN 1 + 2 = 130 1 + 1 = 130 2(1) = 130 (1) = 65 Since 1 = 2 Postulate 3 - 3 Replace CAN with 130 Replace 2 with 1 Combine like terms Divide each side by 2 1 2 T C A The Angle Addition Postulate If AT bisects CAN and mCAN = 130, find 1 and 2. Since AT bisects CAN, 1 = 2. N 1 + 2 = CAN 1 + 2 = 130 1 + 1 = 130 2(1) = 130 (1) = 65 Since 1 = 2, Postulate 3 - 3 Replace CAN with 130 Replace 2 with 1 Combine like terms Divide each side by 2 1 2 T 2 = 65 C A The Angle Addition Postulate