GEOMETRY: TRIANGLES COMMON MISTAKES


 Geoffrey Boone
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1 GEOMETRY: TRIANGLES COMMON MISTAKES 1
2 GeometryClassifying Triangles How Triangles are Classified TypesTriangles are classified by Angles or Sides By Angles Obtuse Trianglestriangles with one obtuse angle. Right Trianglestriangles with only one right angle. Acute Trianglestriangles with all 3 angles acute. Equiangular Triangles triangles with 3 congruent angles(all 3 angles have the same measure. By Sides Scalene Triangles None of the 3 sides are congruent( have the same length) to each other. Isosceles Triangles Two of the 3 sides are congruent to each other. Equilateral Triangles All 3 sides of the triangles are congruent to one another. Confusing the triangles namesusing the names for triangles based on their angles with those based on the sides H R E B S I Incorrect: Triangle BHS is right. Triangle IRA is obtuse. Triangle MEK is equiangular. Correct: Triangle BHS is acute. Triangle IRA is right. Triangle MEK is obtuse. A M K
3 GeometryPythagorean Theorem How to apply the Pythagorean Theorem Pythagorean Theorem states that in a RIGHT triangle, the following lengths have this property Written in geometry, it s A AB + BC = AC 3 C = c B F 15 C Forgetting to take the SQUARE ROOT to find the final length. (i.e. Find the value of c: = c 36 c Incorrect: V Correct: 151 = c 151 = c 151 = c 39 = c )
4 Geometry Proving Triangles Congruent Names of theorems that prove triangles congruent: SSS, SAS, AND ASA For TRIANGLES TO BE CONGRUENT, the 3 sides and 3 angles that go with each other (called corresponding parts) must be congruent, (i.e. have the same measurements), It can be proven that if the following 3 relationships between triangles are true, then the triangle ARE congruent and all 6 facts (3 pairs of congruent sides and 3 pairs of congruent angles) are true. SSS (SideSideSide) If the 3 corresponding sides of one triangle are congruent to the 3 corresponding sides of another triangle, then the triangles are congruent. SAS (SideAngleSide) If two sides and the included angle (the angle formed by these two sides) of one triangle are congruent to the corresponding two sides and included angle of another triangle, then the triangles are congruent. Confusing which corresponding part are congruent which results in picking the wrong theorem. Example: State the theorem that makes these two triangles congruent. B N R 8 W 15 A 8 F ASA (AngleSideAngle) If two angles and the included side (the side between these two angles) of one triangle are congruent to the corresponding two angles and included side of another triangle, then the triangles are congruent. Incorrect: Triangle BNW is congruent to triangle FAR by ASA. Correct: Triangle BNW is congruent to triangle FAR by SSS! 4
5 GeometryAngles continued Types formed by Intersecting lines and Transversals 5 When a transversal line intersects two other lines, parallel or not, names are given to the angles formed based upon the position(s) in which they lie. Corresponding Angles occupy the same position with respect to the intersecting line and transversal (i.e. angles 1 and 5; and 6; 3 and 7; 4 and 8 ) Alternate Interior Angles lie in the interior positions, but on opposite sides of the transversal. (i.e. angles 4 and 6; 3 and 5) Alternate Exterior Angles lie in the exterior positions, also on opposite sides of the transversal. (i.e. angles 1 and 7; and 8) SameSide Interior Angles occupy the interior positions on the sameside of the transversal. (i.e. angles 4 and 5; 3 and 6) SameSide Exterior Angles also occupy the exterior positions but on the sameside of the transversal. (i.e. angles 1 and 8; and 7) Incorrectly naming the angles formed by a transversal. 1 4 Incorrect: Angles and 6 are sameside interior angles. Correct: Angles an 6 are corresponding angles a t b
6 GeometryAngles continued Types defined by their Angle Measurements Acute Angles are angles whose measurements are between 0 and 90 (i.e. Angle A). Mistaking the definitions of angles based upon their measurements. Right Angles are angles whose measurements are 90 (i.e. Angle b). Obtuse Angles are angles whose measurements are between 90 and 180 (i.e. Angle C). Straight Angles are angles whose measurements are 180 (i.e. Angle D). Lines are examples of straight angles. A B C D Incorrect: Angle A is an obtuse angle. Correct: Angle A is an acute angle. 6
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