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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 › Quadrilaterals

    Diagonals of a kite are perpendicular

    Students will be asked to prove the diagonals of a kite are perpendicular. If students can't prove it, show them this proof. Then have students use the theorem to solve some basic problems. Here's an example.

    Conclude by giving your students these challenges:

    • Stars by NRICH
    • Treasure Hunt by NRICH
    • 2007 AMC 8, Problem 2

    Follow the Numbers by NRICH: Order of digits doesn't matter, their initial sum will be the same. For example, 73 and 37 have the same initial sum. In fact, many numbers will have the same initial sum. For example, 4, 13, 22, 31, and 40, all have the same initial sum. From this observation, we can figure out where the numbers 1-100 go by following 18 possibilities, because 1 has the smallest initial sum, which is 1, and 99 has the largest initial sum, which is 18. Some of these 18 will have the same journey, for the reasons previously stated. These are: 1 and 10, 2 and 11, ..., 9 and 18. We also know where a number goes once we find a number we've seen before. Following each journey in turn, remembering that our 1-18 are the initial sums, we have

    • 1, 2, 4, 8, 16, 14, 10, ...
    • 3, 6, 12, ...
    • 5, 10, ...
    • 18, 36, 18, ...

    Thus, every number gets caught in 1 of 3 patterns:

    • 2, 4, 8, 16, 14, 10, 2, ...
    • 6, 12, 6, ...
    • 18, 18, ...

    This strategy can be used to find the patterns for the other two proposed rules as well:

    • Add the digits, then multiply by 3.
    • Add the digits, then multiply by 5.

    Lessons and practice problems