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    • ▾High school geometry
      • ▸Basics of geometry
        • •Proofs in this course
        • •Geometric definitions
        • •Linear pair theorem
        • •Area of a regular polygon
        • •Diagonal length of a right rectangular prism
        • •Distance and midpoint in 3D
        • •Circumcircles and incircles
      • ▸Transformations
        • •Preserved properties of rigid transformations
        • •Find measures using rigid transformations
        • •Find the angle of rotation
        • •Find the sequence of rigid transformations
        • •Reflecting figures
        • •Reflecting a point over any non-horizontal line
      • ▸Congruence
        • •Congruent polygons
        • •Congruent supplements and complements theorems
        • •ASA and AAS congruence
        • •Isosceles triangle theorem and its converse
        • •SSS congruence
        • •SAS congruence
        • •Converse of the Pythagorean theorem
        • •Proving isosceles trapezoids
      • ▾Similarity
        • •Intro to similarity
        • •All circles are similar
        • •Finding angles and side lengths in similar polygons
        • •Finding the scale factor for similar polygons
        • •Third angle theorem
        • •Similar triangles
        • •Ratios of area, surface area, and volume, for similar shapes
        • •Crossed ladders theorem
      • ▸Polygons
        • •Perimeter of composite figures
        • •Finding the volume and surface area of composite figures
        • •Sum of the interior angles of a triangle
        • •Sum of interior angles for simple polygons
        • •Sum of exterior angles
        • •Pentagonal tilings
      • ▸Incenter and circumcenter of a triangle
        • •Intro to the incenter and circumcenter of a triangle
        • •Perpendicular bisector theorem and its converse
        • •Circumcenter theorem
        • •Angle bisector theorem
        • •Incenter theorem
      • ▸Triangles
        • •Euler line
        • •Geometric mean theorems
        • •AM-GM inequality
        • •Triangle angle bisector theorem
        • •Ceva's theorem
        • •Proving the altitude of an isosceles triangle cuts two congruent right triangles
        • •Centroid and orthocenter
        • •Ordering triangle sides and angles
        • •Midsegment and proportionality theorems
        • •Menelaus's theorem
        • •Side splitter theorem
        • •Hinge theorem and its converse
        • •Hypotenuse-leg congruence theorem
        • •Hypotenuse-angle congruence theorem
        • •Areas of triangles and quadrilaterals on grids
        • •Exterior angle theorem
      • ▸Quadrilaterals
        • •Diagonals of a kite are perpendicular
        • •Diagonals of a parallelogram bisect each other
        • •Diagonals of a rhombus are perpendicular bisectors
        • •Area of a rhombus from diagonals
        • •Diagonals of a rectangle are congruent
        • •Opposite sides of a parallelogram are congruent
        • •Opposite angles of a parallelogram are congruent
        • •British flag theorem
        • •Cyclic quadrilaterals
        • •Fuhrmann's theorem
        • •Classifying quadrilaterals from four points
      • ▸Circle theorems
        • •Thales's theorem and its converse
        • •Central angle theorem
        • •Angles standing on the same arc are congruent
        • •Chord properties
        • •Congruent tangents theorem
        • •Radius and tangent theorem
        • •Radii inside right triangles
        • •Alternate segments theorem
        • •Power of a point
        • •Three squares puzzle
      • ▸Volume and surface area
        • •Cavalieri's principle
        • •Volume of prisms and cylinders
        • •Volume and surface area of a sphere
      • ▸Analytic geometry
        • •A square within a square
     › High school geometry › Similarity

    Similar triangles

    First, students will learn how to use the AA similarity theorem, to find missing side lengths for similar triangles.

    Solving similar triangles by Khan Academy

    After that, students will see how the theorem can be used to prove that parallel lines always have the same slope.

    Proof: parallel lines have the same slope by Khan Academy

    Next, students will see how the theorem can be used to prove that perpendicular lines always have negative reciprocal slope.

    Proof: perpendicular lines have opposite reciprocal slopes by Khan Academy

    After that, students will see how the Pythagorean theorem can be proven using similar triangles.

    Pythagoras Theorem: Proof #2: Using 3 similar triangles by Joel Speranza Math

    Next, give your students these challenges.

    Conclude by giving your students these challenges:

    • Overturning Fracsum by NRICH
    • Odds and Evens by NRICH
    • Only Connect by NRICH
    • Maximaze / Minimaze by Pierce School: Problem / Solution

    Additional lessons and practice problems