HW 4.1 PAP key-1

Triangle Congruence Statements

Congruence Problems

  • Complete the congruence statements based on given pairs:

      1. LC III ?BK

      1. KJ III ?CM

      1. LK III ?LC

      1. LM III ?L.J HY

      1. ACML III ?AKJB

      1. AKBJ III ?CCLM

List Congruent Sides and Angles

  • Find four pairs of congruent sides:

      1. Pair 1: [Specify pairs based on the triangle]

      1. Pair 2:

      1. Pair 3:

      1. Pair 4:

  • Find four pairs of congruent angles:

      1. Angle 1: [Specify angles based on the triangle]

      1. Angle 2:

      1. Angle 3:

      1. Angle 4:

Questions on Triangle Congruence

  • For Exercises 19 and 20, conclude if the triangles are congruent and justify:

      1. ATRK and ATUK: YES

      • Reason: All corresponding sides are equal (must specify which sides).

      1. ASPQ and ATUV: NO

      • Reason: Only one segment corresponds between triangles.

Proving Congruence with the Given Properties

Definitions and Proof Statements

  • Given: AB || DC, angles LBAC and LDCA are supplementary.

Proof Steps

  1. Statement: AB || DC

    • Reason: Given.

  2. Angles: LBAC and LDCA are alternative interior angles.

    • Reason: Definition of alternate interior angles.

  3. Properties: LBAC || LDCA, confirming angle relationships.

    • Reason: Alternate Interior Angles Theorem.

  4. Segment Equality: BC III AD

    • Reason: Given relationships define congruence.

  5. Conclude: AABC III ACDA

    • Reason: Definition of Congruent Triangles.

Connect Mathematical Ideas

Angle Measurements and Line Segments

  • Find measures and lengths using provided equations:

      1. If mLA = x + 10, mLD = 2x:

      • Solve: mLA = 20, mLB = 12.

      1. Given m/B = 3y and mLE = 6y - 12:

      • Determine values: y = 4.

      1. For sides BC = 3z + 2, EF = z + 6, find equivalency:

      • z = 2, BC = 8, EF = 8.

      1. Compare AC = 7a + 5 with DF = 5a + 9:

      • Solve: a = 2, AC = 19, DF = 19.

      1. If ADEF III ALMN, consider correct congruence statements:

      • Options:

        • A: DE III LN

        • B: FE III NL

        • C: LN = FF

        • D: LM = FF

Variable Values Calculation

  • Solve for variables in equations:

      1. For 3x = 45, find:

      • x = 15.

      1. For 6x = 30, solve:

      • x = 5.

      1. Given equations: 2t = 4,

      • Solve for t: t = 2.

Further Proofs and Theorems

  • Given the relationships among lines and angles:

    • Objective: Prove APRS III AQTS with alternate interior angles.

    • Step 1: Identify properties of bisected angles and segments.

    • Step 2: Use graphical interpretations of segments involving equal parts and angles for proving congruence using alternate interior angles theorem.