Angle bisectors

Solving Equations

  • Equation: 2x+1=14x−372x + 1 = 14x - 37

  • Combine like terms: 2x−14x=−37−12x - 14x = -37 - 1

  • Simplify: −12x=−38-12x = -38

  • Solve for x: x=−38−12=196x = \frac{-38}{-12} = \frac{19}{6}

Finding Angle Measures

  • Given: 5x−115x - 11 and 3x+173x + 17 are angle measures.

  • Equation: 5x−11=3x+175x - 11 = 3x + 17

  • Combine like terms: 5x−3x=17+115x - 3x = 17 + 11

  • Simplify: 2x=282x = 28

  • Solve for x: x=14x = 14

  • Angle measure: 5(14)−11=70−11=59∘5(14) - 11 = 70 - 11 = 59^{\circ}

  • Supplementary angle: 59×2=118∘59 \times 2 = 118^{\circ}

Solving for x

  • Equation: 6x−26=3x+346x - 26 = 3x + 34

  • Combine like terms: 6x−3x=34+266x - 3x = 34 + 26

  • Simplify: 3x=603x = 60

  • Solve for x: x=20x = 20

  • Check: 6(20)−26=120−26=94∘6(20) - 26 = 120 - 26 = 94^{\circ}

Angle Bisector Theorem

  • Definition: If a point is on the bisector of an angle, then the point is equidistant from the sides of the angle.

  • Given: AD‾\overline{AD} bisects ∠BAC\angle BAC, BD‾⊥AB‾\overline{BD} \perp \overline{AB}, and DC‾⊥AC‾\overline{DC} \perp \overline{AC}

  • Conclusion: BD‾≅DC‾\overline{BD} \cong \overline{DC}

Converse of the Angle Bisector Theorem

  • Definition: If a point is in the interior of an angle and equidistant from the sides of the angle, then the point is on the angle bisector.

  • Given: BD‾≅DC‾\overline{BD} \cong \overline{DC}, AB‾⊥BD‾\overline{AB} \perp \overline{BD}, and AC‾⊥CD‾\overline{AC} \perp \overline{CD}

  • Conclusion: ∠BAD≅∠DAC\angle BAD \cong \angle DAC, and AD‾\overline{AD} bisects ∠BAC\angle BAC

Example Problem

  • Equation: 4x+30=9x−54x + 30 = 9x - 5

  • Combine like terms: 4x−9x=−5−304x - 9x = -5 - 30

  • Simplify: −5x=−35-5x = -35

  • Solve for x: x=7x = 7

Homework 2: Perpendicular & Angle Bisectors

  1. If RT bisects SU, find each measure.

    • Given: ST = 23, RU = 4x - 19, VU = 6x - 13

    • Solve for x: 4x−19=6x−13⇒−2x=6⇒x=−34x - 19 = 6x - 13 \Rightarrow -2x = 6 \Rightarrow x = -3

    • RU = 4(−3)−19=−12−19=−314(-3) - 19 = -12 - 19 = -31

    • VU = 6(−3)−13=−18−13=−316(-3) - 13 = -18 - 13 = -31

    • SV = 5x + 17

    • RV = 8x - 37

    • Solve for x: 8x-37=5x+17 => 3x=54 => x=18

    • SV = 5(18) + 17 = 90 + 17 = 107

    • RV = 8(18) - 37 = 144 - 37 = 107

    • SU = ST + TU= 23 + 23 = 46
      Solve for x: 4x+10=6x−16⇒−2x=−26⇒x=134x + 10 = 6x - 16 \Rightarrow -2x = -26 \Rightarrow x = 13

    • MN = 4(13)+10=52+10=624(13) + 10 = 52 + 10 = 62

  2. Solve for x

    • Solve for x: 9x-13=7x-1 =>2x = 1 => x = 7

    • CD = 7(7)−1=49−1=487(7) - 1 = 49-1 = 48

  3. Solve for x: 3x+4=5x−16⇒−2x=−20⇒x=103x + 4 = 5x - 16 \Rightarrow -2x = -20 \Rightarrow x = 10

    • JL = 3(10)+4=343(10) + 4 = 34

    • Solve for x: 12x−20=3412x - 20 = 34; x = unknown

    • Solve for x: 34*2=68

  4. If QT is the perpendicular bisector of PR, find each measure.

    • Given: PQ = 5y - 31, QR = 2y + 5, PT = 4x + 4, TR = 7x - 17

    • Solve for y: 5y−31=2y+5⇒3y=36⇒y=125y - 31 = 2y + 5 \Rightarrow 3y = 36 \Rightarrow y = 12

    • PQ = 5(12)−31=60−31=295(12) - 31 = 60 - 31 = 29

    • QR = 2(12)+5=24+5=292(12) + 5 = 24 + 5 = 29

    • Solve for x: 4x+4=7x−17⇒−3x=−21⇒x=74x + 4 = 7x - 17 \Rightarrow -3x = -21 \Rightarrow x = 7

    • PT = 4(7)+4=28+4=324(7) + 4 = 28 + 4 = 32

    • TR = 7(7)−17=49−17=327(7) - 17 = 49 - 17 = 32

    • PS = 32

    • SR = 32

    • PR = 32 + 32= 64

    • PT = 32