Unit 4: Congruent Triangles Study Guide

4.1: Apply Triangle Sum Properties

  • Definition of a Triangle: A triangle is a polygon with three sides and three vertices.

  • Classifying Triangles by Sides:

    • Scalene Triangle: No congruent sides.
    • Isosceles Triangle: At least 2 congruent sides.
    • Equilateral Triangle: 3 congruent sides.

  • Classifying Triangles by Angles:

    • Acute Triangle: 3 acute angles (all angles are less than 90∘90^\circ).
    • Right Triangle: 1 right angle (90∘90^\circ).
    • Obtuse Triangle: 1 obtuse angle (greater than 90∘90^\circ).
    • Equiangular Triangle: 3 congruent angles (each is 60∘60^\circ).

  • Example 1: Coordinate Classification:

    • Given vertices: P(−1,2)P(-1, 2), O(0,0)O(0, 0), and Q(6,3)Q(6, 3).
    • Distance Formula Calculations:
      • OP=(−1−0)2+(2−0)2=1+4=5OP = \sqrt{(-1 - 0)^2 + (2 - 0)^2} = \sqrt{1 + 4} = \sqrt{5}
      • OQ=(6−0)2+(3−0)2=36+9=45=35OQ = \sqrt{(6 - 0)^2 + (3 - 0)^2} = \sqrt{36 + 9} = \sqrt{45} = 3\sqrt{5}
      • PQ=(6−(−1))2+(3−2)2=72+12=49+1=50=52PQ = \sqrt{(6 - (-1))^2 + (3 - 2)^2} = \sqrt{7^2 + 1^2} = \sqrt{49 + 1} = \sqrt{50} = 5\sqrt{2}
    • Result: Since all sides are different lengths, △PQO\triangle PQO is Scalene.
    • Right Triangle Verification (Slopes):
      • Slope of OQ=3−06−0=36=12OQ = \frac{3-0}{6-0} = \frac{3}{6} = \frac{1}{2}
      • Slope of OP=2−0−1−0=−2OP = \frac{2-0}{-1-0} = -2
      • Since the slopes are negative reciprocals (12×−2=−1\frac{1}{2} \times -2 = -1), the lines are perpendicular (OQ⊥OPOQ \perp OP).
    • Final Classification: △PQO\triangle PQO is a Right Scalene Triangle.

  • Triangle Angle Theorems:

    • THEOREM 4.1: Triangle Sum Theorem: The sum of the measures of the interior angles of a triangle is 180∘180^\circ.
      • Formula: m∠A+m∠B+m∠C=180∘m\angle A + m\angle B + m\angle C = 180^\circ.
    • THEOREM 4.2: Exterior Angle Theorem: The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles.
      • Formula: m∠1=m∠A+m∠Bm\angle 1 = m\angle A + m\angle B.
    • COROLLARY to the Triangle Sum Theorem: The acute angles of a right triangle are complementary.
      • Formula: m∠A+m∠B=90∘m\angle A + m\angle B = 90^\circ.

  • Sample Proofs and Algebra Applications:

    • Example 4: Find m∠JKMm\angle JKM.
      • Given interior angles 70∘70^\circ and x∘x^\circ, and an exterior angle (2x−5)∘(2x - 5)^\circ.
      • By Exterior Angle Theorem: 70+x=2x−570 + x = 2x - 5
      • Solving for xx: x=75x = 75
      • Substitute into expression: m∠JKM=2(75)−5=145∘m\angle JKM = 2(75) - 5 = 145^\circ.
    • Example 5: Find the measure of each angle in a right triangle where acute angles are xx and 2x2x.
      • 2x+x=90→3x=90→x=302x + x = 90 \rightarrow 3x = 90 \rightarrow x = 30.
      • Angles are 30∘30^\circ and 60∘60^\circ.

4.2: Apply Congruence and Triangles

  • Definition of Congruent: Figures that have the exact same size and shape are congruent.

  • Congruence Statements: When writing a congruence statement (e.g., △ABC≅△FED\triangle ABC \cong \triangle FED), corresponding vertices must be listed in the same order.

    • Corresponding Angles: ∠A≅∠F\angle A \cong \angle F, ∠B≅∠E\angle B \cong \angle E, ∠C≅∠D\angle C \cong \angle D.
    • Corresponding Sides: AB‾≅FE‾\overline{AB} \cong \overline{FE}, BC‾≅ED‾\overline{BC} \cong \overline{ED}, AC‾≅FD‾\overline{AC} \cong \overline{FD}.

  • Theorems of Congruence:

    • THEOREM 4.3: Third Angles Theorem: If two angles of one triangle are congruent to two angles of another triangle, then the third angles are also congruent.
    • THEOREM 4.4: Properties of Congruent Triangles:
      • Reflexive: △ABC≅△ABC\triangle ABC \cong \triangle ABC
      • Symmetric: If △ABC≅△DEF\triangle ABC \cong \triangle DEF, then △DEF≅△ABC\triangle DEF \cong \triangle ABC
      • Transitive: If △ABC≅△DEF\triangle ABC \cong \triangle DEF and △DEF≅△JKL\triangle DEF \cong \triangle JKL, then △ABC≅△JKL\triangle ABC \cong \triangle JKL

  • Example 2: Algebra with Congruent Polygons:

    • Given Quad DEFG≅DEFG \cong Quad SPQRSPQR.
    • Side FGFG corresponds to QRQR: 2x−4=12→2x=16→x=82x - 4 = 12 \rightarrow 2x = 16 \rightarrow x = 8.
    • Angle QQ corresponds to angle FF (106∘106^\circ). Note: transcript handwriting sets 6y+x6y + x to 6868, resulting in y=10y = 10.

4.3 - 4.5: Proving Triangle Congruence

  • Five Main Postulates/Theorems:

    1. SSS (Side-Side-Side): All three sides are congruent.
    2. SAS (Side-Angle-Side): Two sides and the included angle (the angle between the sides) are congruent.
    3. ASA (Angle-Side-Angle): Two angles and the included side are congruent.
    4. AAS (Angle-Angle-Side): Two angles and a non-included side are congruent.
    5. HL (Hypotenuse-Leg): Used for Right Triangles only; hypotenuse and one leg must be congruent.

  • Important Warning: SSA (Side-Side-Angle) is NOT a valid congruence postulate.

  • Example 6: Determining Sufficient Information:

    • Identifying if SAS can be used for △LMN\triangle LMN and △NQP\triangle NQP: Yes, if the vertical angles at point NN are used.
    • Identifying if SAS can be used for △QRV\triangle QRV and △TSU\triangle TSU: No.

  • Example 8: Analyzing Postulates:

    • a) Given ∠A≅∠D\angle A \cong \angle D, AB=DEAB = DE, AC=DFAC = DF: Valid by SAS.
    • b) Given ∠B≅∠E\angle B \cong \angle E, ∠C≅∠F\angle C \cong \angle F, AC=DEAC = DE: Valid by AAS.

4.6: Use Congruent Triangles (CPCTC)

  • CPCTC: Corresponding Parts of Congruent Triangles are Congruent. This is used in proofs after proving triangles are congruent.

  • Example 1 Proof Implementation:

    • Given: ∠1≅∠2\angle 1 \cong \angle 2, ∠RTQ≅∠RTS\angle RTQ \cong \angle RTS.
    • Prove: QT=STQT = ST.
    • Steps:
      1. RT‾≅RT‾\overline{RT} \cong \overline{RT} (Reflexive Property).
      2. △RTS≅△RTQ\triangle RTS \cong \triangle RTQ by AAS Postulate.
      3. QT=STQT = ST by CPCTC.

4.7: Isosceles and Equilateral Triangles

  • Anatomy of an Isosceles Triangle:

    • Legs: The two congruent sides.
    • Base: The non-congruent third side.
    • Vertex Angle: The angle formed by the legs.
    • Base Angles: The angles adjacent to the base.

  • Theorems:

    • THEOREM 4.7: Base Angles Theorem: If two sides of a triangle are congruent, then the angles opposite them are congruent (Base angles are congruent).
    • THEOREM 4.8: Converse of Base Angles Theorem: If two angles of a triangle are congruent, then the sides opposite them are congruent.

  • Corollaries:

    • A triangle is equilateral if and only if it is equiangular. Each angle measure in an equilateral triangle is exactly 60∘60^\circ.

  • Example 5: Algebra with Equilateral/Isosceles Triangles:

    • In an equilateral triangle with side expressions: x+1=4→x=3x + 1 = 4 \rightarrow x = 3.
    • Linear pair calculation: If the inner angle of an equilateral triangle is 60∘60^\circ, the supplementary exterior angle y=120∘y = 120^\circ.

4.8: Perform Congruence Transformations

  • Key Definitions:

    • Transformation: An operation that moves or changes a geometric figure.
    • Image: The new figure resulting from a transformation.
    • Rigid Transformation: A move that preserves both size and shape (includes translations, reflections, and rotations).

  • Summary of Transformations:

    • Translation: Slide of a figure in a specific direction.
      • Coordinate Notation: (x,y)→(x+a,y+b)(x, y) \rightarrow (x + a, y + b).
    • Reflection: Flip over a line (line of reflection).
      • Over xx-axis: (x,y)→(x,−y)(x, y) \rightarrow (x, -y).
      • Over yy-axis: (x,y)→(−x,y)(x, y) \rightarrow (-x, y).
    • Rotation: Turn around a fixed center point (origin).
      • Turns preserve distance from the center of rotation.

  • Example 4: Rotation Check:

    • Given segments ABAB and CDCD. To check if CDCD is a rotation of ABAB, verify if the corresponding points (A→CA \rightarrow C, B→DB \rightarrow D) form the same angle relative to the origin and maintain congruent distances from the origin. In this example, the figure demonstrated a 90∘90^\circ clockwise rotation.