You can prove two triangles are identical using SSS, SAS, ASA or AAS, and then claim every matching part is equal. You can also spot when the given information simply is not enough.
Mark the shared side and the vertical angles before you choose a rule.
Write a two-column proof that ends with matching parts being equal.
Explain why two sides and a non-included angle do not prove congruence.
Worked example
Triangles ABC and ADC share side AC. AB = AD = 6 and BC = DC = 8. Are they congruent?
AC = AC, because it is literally the same segment
Three pairs of sides now match: 6 = 6, 8 = 8, AC = AC
AnswerYes, congruent by SSS
Most common mistakeClaiming congruence from three equal angles: AAA only fixes the shape, so a 3-4-5 triangle and a 6-8-10 triangle would be wrongly called identical.
You can decide whether two figures are the same shape at different sizes and use the scale factor to find missing lengths. You can also scale an area correctly instead of guessing.
Build a proportion from the matching sides of two similar triangles.
Find the scale factor between a 5 cm side and its 12.5 cm partner.
Explain why doubling every side multiplies the area by four.
Worked example
Two similar triangles have matching sides 5 and 12.5. Another side of the small one is 8. Find its partner.
Scale factor = 12.5 / 5 = 2.5
8 x 2.5
Answer20
Most common mistakeAdding the difference instead of multiplying: since 12.5 - 5 = 7.5, answering 8 + 7.5 = 15.5 rather than 20.
You can find a missing side with the Pythagorean theorem and a missing angle with sine, cosine or tangent. You can also solve ladder, ramp and shadow word problems.
Find the hypotenuse of a triangle with legs 9 and 12.
Choose tangent when the problem gives you the opposite and adjacent sides.
Use the 45-45-90 shortcut to get the hypotenuse from one leg.
Worked example
A ladder leans against a wall. It reaches 12 ft up and its foot sits 5 ft out. How long is the ladder?
5 squared + 12 squared = 25 + 144 = 169
The square root of 169 is 13
Answer13 ft
Most common mistakeAdding the legs instead of their squares: answering 5 + 12 = 17 ft, longer than the ladder can possibly be.
You can find arc length, sector area and the angles made by chords, tangents and inscribed angles. You can also write a circle's equation from its centre and radius.
Use the rule that an inscribed angle is half the arc it sits on.
Find the arc length cut by a 90 degree central angle in a circle of radius 6.
Write the equation of the circle with centre (2, -3) and radius 5.
Worked example
An inscribed angle sits on an arc of 130 degrees. How big is the angle?
An inscribed angle is half its arc
130 / 2
Answer65 degrees
Most common mistakeUsing the arc measure as the angle: answering 130 degrees, which is the central angle standing on the same arc, not the inscribed one.
You can find surface area and volume for prisms, cylinders, cones, pyramids and spheres. You can also work backwards from a volume to a missing height or radius.
Find the volume of a cylinder with radius 3 and height 10.
Apply the one-third rule that separates cones and pyramids from their full-height partners.
Find a cube's edge length when its volume is 216.
Worked example
Find the volume of a cone with radius 3 cm and height 7 cm
V = (1/3) x pi x r squared x h
V = (1/3) x pi x 9 x 7 = 21 pi
AnswerAbout 65.97 cubic cm
Most common mistakeUsing the cylinder formula on a cone: reporting 63 pi, about 197.9 cubic cm, which is three times too big.
You can put a shape on the grid and prove facts about it with the distance, midpoint and slope formulas. You can also show two lines are parallel or perpendicular using numbers instead of a picture.
Show two sides are parallel by getting the same slope for both.
Use the midpoint formula to prove a parallelogram's diagonals cut each other in half.
Prove a triangle has a right angle by showing two slopes multiply to -1.
Worked example
Is the triangle A(0,0), B(4,2), C(2,-4) right-angled at A?
Slope of AB = 2/4 = 1/2
Slope of AC = -4/2 = -2
Answer(1/2) x (-2) = -1, so yes, the angle at A is 90 degrees
Most common mistakeTesting for equal slopes instead of a product of -1: since 1/2 is not -2, the right angle gets missed completely.
Before timing yourself, check whether you can explain Transformations from a blank page. Then connect it to Triangle congruence. If either explanation depends on copying a formula, review the unit first and complete two untimed examples.
Use tools to verify, not to choose the method for you
The Geometry calculator can test calculations and representations used in High School Geometry. Make the setup yourself, predict the sign or scale, and compare the tool result with that prediction. Use the formula library to check conditions as well as notation.
Know when to move to the full test
Move from High School Geometry practice to the complete course test after you can correct a missed problem without reopening the worked answer. Record the earliest wrong decision—not only the final score—so the next study session has a precise target.
Before and after the syllabus
Learn the ideas, then practise them
The unit list tells you what is covered. These two pages are where the method is explained and where you find out whether it stuck.
Start with Transformations if you are following the full sequence. If that unit feels automatic, open the High School Geometry practice page, choose mixed review, and let the first errors identify the earliest prerequisite to revisit.
How do I know I am ready for the course test?
You are ready when you can choose a method without a hint, show the governing steps, and explain why the result is reasonable. Use the complete High School Geometry test only after you can correct practice errors from a blank page.
Which calculator supports this course?
The Geometry calculator supports the calculations and representations used in this course. Use it to test or visualize a result after making your own setup.
