Triangles: SPI Mathematics

Formulas (4)

• The angle sum of a triangle · Angle sum
  $$\angle A + \angle B + \angle C = 180^\circ \qquad ak + bk + ck = 180^\circ$$
• The exterior angle of a triangle · Exterior angle property
  $$\text{exterior angle} = \text{sum of the two interior opposite angles}$$
• Any two sides are longer than the third · Triangle inequality
  $$a + b > c \qquad |a - b| < c < a + b$$
• Pythagoras theorem and its converse · Pythagoras theorem
  $$\text{hypotenuse}^2 = \text{base}^2 + \text{height}^2$$

Watch out for (6)

• Dividing 180 by the wrong number
  → The angle sum of a triangle
• Which angles of an isosceles triangle are equal
  → The angle sum of a triangle
• Taking 180° away instead of the given angle
  → The exterior angle of a triangle
• Equal is not enough
  → Any two sides are longer than the third
• Adding when you should subtract
  → Pythagoras theorem and its converse
• Testing the wrong side as the hypotenuse
  → Pythagoras theorem and its converse

Formulas (1)

• Congruent triangles and the order of the letters · Matching parts
  $$\triangle ABC \cong \triangle PQR \Rightarrow AB = PQ,\ BC = QR,\ CA = RP,\ \angle A = \angle P,\ \angle B = \angle Q,\ \angle C = \angle R$$

Reference tables (1)

Watch out for (4)

• Matching letters in alphabetical order
  → Congruent triangles and the order of the letters
• Equal angles alone are not congruence
  → Congruent triangles and the order of the letters
• Calling RHS when the side given is not the hypotenuse
  → The congruence rules SSS, SAS, ASA, AAS and RHS
• An angle that is not between the sides
  → The included angle and the included side

Formulas (3)

• Corresponding sides are in one ratio · Ratio of corresponding sides
  $$\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD} \qquad \frac{\text{height}_1}{\text{shadow}_1} = \frac{\text{height}_2}{\text{shadow}_2}$$
• Basic proportionality theorem · Basic proportionality theorem
  $$DE \parallel BC \Rightarrow \frac{AD}{DB} = \frac{AE}{EC} \quad \text{and} \quad \frac{AD}{AB} = \frac{AE}{AC}$$
• The angle bisector divides the opposite side · Angle bisector theorem
  $$\frac{BD}{DC} = \frac{AB}{AC}$$

Reference tables (1)

Watch out for (6)

• Finding the angle at the wrong corner
  → When two triangles are similar
• Pairing sides by their position on the page
  → Corresponding sides are in one ratio
• Turning the ratio upside down
  → Corresponding sides are in one ratio
• Mixing a part with a whole
  → Basic proportionality theorem
• Joining B to the wrong point
  → Basic proportionality theorem
• Pairing a part with the far side
  → The angle bisector divides the opposite side

Formulas (2)

• Triangles with the same height · Area of a triangle
  $$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} \qquad \frac{\text{ar}_1}{\text{ar}_2} = \frac{\text{base}_1}{\text{base}_2} \text{ (same height)}$$
• Areas of similar triangles · Area ratio of similar triangles
  $$\frac{\text{ar}(\triangle ABC)}{\text{ar}(\triangle DEF)} = \left(\frac{AB}{DE}\right)^2 = \left(\frac{\text{height}_1}{\text{height}_2}\right)^2$$

Watch out for (4)

• Squaring a ratio that should not be squared
  → Triangles with the same height
• Giving the side ratio as the area ratio
  → Areas of similar triangles
• Forgetting the square root on the way back
  → Areas of similar triangles
• Stopping at the whole triangle
  → Areas of similar triangles