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3.2 – Congruent Right Triangles

Objectives

  • Explore the right triangle congruence shortcut theorems — HL, LL, HA, and LA.
  • Discover the relationship between right triangle congruence theorems and the congruence theorems for non-right triangles.
  • Use the perpendicular bisector theorem to find unknown side lengths or determine if two right triangles are congruent.
  • Use the angle bisector theorem to find unknown angle measures in right triangles.

 

Key Terms

  • HA Congruence Theorem – “Hypotenuse – (acute) Angle”
    • A theorem stating that if the hypotenuse and an acute angle of one right triangle are congruent to the hypotenuse and corresponding acute angle of another right triangle, then the right triangles are congruent.
    • Special case of AAS – right angle, acute angle, hypotenuse
  • HL congruence theorem – “Hypotenuse – Leg”
    • A theorem stating that if the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and corresponding leg of another right triangle, then the right triangles are congruent.
  • LA Congruence Theorem – “Leg – (acute) Angle”
    • A theorem stating that if the leg and an acute angle of one right triangle are congruent to the leg and a corresponding acute angle of another right triangle, then the triangles are congruent.
    • Special case of ASA – acute angle, leg, right angle or right angle, leg, acute angle
      • Leg is included between angles.
    • Special case of AAS – acute angle, right angle, leg
      • Leg is not included between angles.
  • LL Congruence Theorem – “Leg – Leg”
    • A theorem stating that if the legs of one right triangle are congruent to the corresponding legs of another right triangle, then the right triangles are congruent.
    • Special case of SSS – Leg, leg, hypotenuse (use the Pythagorean theorem to find the missing hypotenuse, resulting in knowing the lengths and congruence of all 3 sides).
    • Special case of SAS – Leg, right angle, leg (with the right angle included between the legs of these triangles).

 

Notes

Congruence Shortcuts for Right Triangles
GeoA 3.02 - Right Angle Theorems GeoA 3.02 - Congruence Chart
Examples
  • Leg – Angle (acute)

GeoA 3.02 - LA (non-incl) AAS


GeoA 3.02 - LA (incl) ASA

  • Hypotenuse – Angle (acute)

 

GeoA 3.02 - HA AAS

Proving Theorems with Congruent Right Triangles
  • Congruent right triangles are used to prove the following theorems:
    • Perpendicular bisector theorem
    • Perpendicular bisector theorem converse
    • Angle bisector theorem
    • Angle bisector theorem converse

GeoA 3.02 - Bisector Theorems

  • Rule: the distance from a point to a line is always measured along the perpendicular segment that joins them.

GeoA 3.02 - Distance from Pt to Line

  • Perpendicular Bisector Theorem – If a point (C) is on the perpendicular bisector (blue) of a segment (red), then it is equidistant from the endpoints of the segment (red).
    • The blue line crosses the red line segment at 90°
    • The red line segment has been bisected (cut in half)
      • Both halves are congruent to one another
    • The grey line segments connect point (C) to the red line segment’s endpoints
      • They are congruent to one another
    • The distance from C to D can be found using the Pythagorean theorem
      • \overline{CD} \cong \overline{CD} – Reflexive property

GeoA 3.02 - Perpendicular Segment


  • Proof of the Perpendicular Bisector Theorem
    • Copy the steps to the following proof on Apex Study 3.2 Pg 23 (or from the animation below)
    • Prove: C is equidistant from the endpoints of \overline{AB}

GeoA 3.02 - Proof Perp Bisector

  • Angle Bisector Theorem: If a point is on the bisector (line segment) of an angle, then that point is equidistant from the two sides of the angle.
    • Prove: D lies on the angle bisector of \angle BAC

GeoA 3.02 - Angle Bisector

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