This math brainteaser challenges you to find a simple, elegant solution to a seemingly complex problem! Students will use geometry principles and their knowledge about triangles to solve this puzzle. Can you figure it out?

Content Standard(s):

Mathematics MA2015 (2016) Grade: 9-12 Geometry

10 ) Prove theorems about triangles. Theorems include measures of interior angles of a triangle sum to 180^{o}, base angles of isosceles triangles are congruent, the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length, and the medians of a triangle meet at a point. [G-CO10]

Alabama Alternate Achievement Standards

AAS Standard: M.G.AAS.HS.10- Given a measure of a leg or base angle of an isosceles triangle, identify the measure of the other leg or other base angle.

Mathematics MA2015 (2016) Grade: 9-12 Geometry

16 ) Use the properties of similarity transformations to establish the angle-angle (AA) criterion for two triangles to be similar. [G-SRT3]

Mathematics MA2015 (2016) Grade: 9-12 Geometry

17 ) Prove theorems about triangles. Theorems include a line parallel to one side of a triangle divides the other two proportionally, and conversely; and the Pythagorean Theorem proved using triangle similarity. [G-SRT4]

Mathematics MA2019 (2019) Grade: 9-12 Geometry with Data Analysis

32. Use coordinates to prove simple geometric theorems algebraically.

Unpacked Content

Evidence Of Student Attainment:

Students:

Given coordinates and geometric theorems and statements defined on a coordinate system, use the coordinate system and logical reasoning to justify (or deny) the statement or theorem, and to critique arguments presented by others.

Teacher Vocabulary:

Simple geometric theorems

Simple geometric figures

Knowledge:

Students know:

Relationships (e.g. distance, slope of line) between sets of points.

Properties of geometric shapes.

Coordinate graphing rules and techniques.

Techniques for presenting a proof of geometric theorems.

Skills:

Students are able to:

Accurately determine what information is needed to prove or disprove a statement or theorem.

Accurately find the needed information and explain and justify conclusions.

Communicate logical reasoning in a systematic way to present a mathematical proof of geometric theorems.

Understanding:

Students understand that:

Modeling geometric figures or relationships on a coordinate graph assists in determining truth of a statement or theorem.

Geometric theorems may be proven or disproven by examining the properties of the geometric shapes in the theorem through the use of appropriate algebraic techniques.

Diverse Learning Needs:

Essential Skills:

Learning Objectives: GEO.32.1: Apply formulas, and properties of polygons, angles, and lines to draw conclusions from the given information.
GEO.32.2: Identify properties of perpendicular and parallel lines, properties of polygons.
GEO.32.3: Illustrate polygons created by given coordinates on a coordinate plane.
GEO.32.4: Identify distance formula, circle formula, Pythagorean Theorem, midpoint.

Prior Knowledge Skills:

Define quadrant, coordinate plane, coordinate axes (x-axis and y-axis), horizontal, vertical, and reflection.

Demonstrate an understanding of an extended coordinate plane.

Draw and label a 4 quadrant coordinate plane.

Draw and extend vertical and horizontal number lines.

Interpret graphing points in all four quadrants of the coordinate plane in real-world situations.

Recall how to graph points in all four quadrants of the coordinate plane.

Define ordered pairs.

Name the pairs of integers and/or rational numbers of a point on a coordinate plane.

Demonstrate when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.

Identify which signs indicate the location of a point in a coordinate plane.

Recall how to plot ordered pairs on a coordinate plane.

Identify the length between vertices on a coordinate plane.

Calculate the perimeter and area using the distance between the vertices.

Define a right angle, Pythagorean Theorem, converse, and proof.

Recognize examples of right triangles.

Demonstrate how to find square roots.

Solve problems with exponents.

Alabama Alternate Achievement Standards

AAS Standard: M.G.AAS.10.31a When given an isosceles triangle and a measure of a leg or base angle, identify the measure of the other leg or base angle.
M.G.AAS.10.31b When given a parallelogram and the measure of one side or one angle, identify the measure of the opposite side or angle.

Mathematics MA2019 (2019) Grade: 9-12 Geometry with Data Analysis

34. Use congruence and similarity criteria for triangles to solve problems in real-world contexts.

Unpacked Content

Evidence Of Student Attainment:

Students:
Given a contextual situation involving triangles,

Determine solutions to problems by applying congruence and similarity criteria for triangles to assist in solving the problem.

Justify solutions and critique the solutions of others.

Given a geometric figure, establish and justify relationships in the figure through the use of congruence and similarity criteria for triangles

Teacher Vocabulary:

Congruence and similarity criteria for triangles

Knowledge:

Students know:

Criteria for congruent (SAS, ASA, AAS, SSS) and similar (AA) triangles and transformation criteria.

Techniques to apply criteria of congruent and similar triangles for solving a contextual problem.

Techniques for applying rigid motions and dilations to solve congruence and similarity problems in real-world contexts.

Skills:

Students are able to:

Accurately solve a contextual problem by applying the criteria of congruent and similar triangles.

Provide justification for the solution process.

Analyze the solutions of others and explain why their solutions are valid or invalid.

Justify relationships in geometric figures through the use of congruent and similar triangles.

Understanding:

Students understand that:

Congruence and similarity criteria for triangles may be used to find solutions of contextual problems.

Relationships in geometric figures may be proven through the use of congruent and similar triangles.

Diverse Learning Needs:

Essential Skills:

Learning Objectives: GEO.34.1: Develop an equation from given information to prove congruence or similarity.
GEO.34.2: Illustrate congruence and similarity in geometric figures.
GEO.34.3: Apply proportional reasoning to real-world scenarios.
GEO.34.4: Solve proportions.

Prior Knowledge Skills:

Analyze an image and its dilation to determine if the two figures are similar.

Identify similar figures.

Define similar.

Identify congruent figures.

Identify attributes of two-dimensional figures.

Compare rotations to translations.

Compare reflections to rotations.

Compare translations to reflections.

Define congruent and sequence.

Apply the rule of proportional relationship to real-world context.

Recognize similar triangles.

Define similar triangles, intercept, slope, vertical, horizontal, and origin.

Demonstrate how to plot points on a coordinate plane using ordered pairs from table.

Analyze the graph to determine the rate of change.

Generate the slope of a line using given ordered pairs.

Graph a function given the slope-intercept form of an equation.

Identify the slope-intercept form (y=mx+b) of an equation where m is the slope and y is the y-intercept.

Graph a linear equation given the slope-intercept form of an equation.

Recognize that two sets of points with the same slope may have different y-intercepts.

Identify the slope-intercept form (y=mx+b) of an equation where m is the slope and y is the y-intercept.

Recall that for a relationship to be proportional, the graph must pass through the origin.

Demonstrate how to graph on a Cartesian plane.

Recall that for a relationship to be proportional, both variables must start at zero.

Identify the unit rate of two quantities.

Recall how to write a ratio of two quantities as a fraction.

Recall equivalent ratios and origin on a coordinate (Cartesian) plane.

Define proportional, independent variable, dependent variable, and unit rate.

Identify proportional relationships.

Locate/use scale on a map.

Define scale, scale drawings, length, area, and geometric figures.

Use a table or graph to determine whether two quantities are proportional.

Define equivalent ratios and origin.

Define unit rate, proportions, area, length, and ratio.

Recognize polygons.
M. 6.3.4: Restate real-world problems or mathematical problems.
M. 6.3.3: Calculate unit rate or rate by using ratios or proportions.
M. 6.3.2: Create a ratio or proportion from a given word problem, diagram, table, or equation.
M. 6.3.1: Define ratio, rate, proportion, percent, equivalent, input, output, ordered pairs, diagram, unit rate, and table.
M. 6.3.16: Form a ratio.
M. 6.3.15: Solve a proportion using part over whole equals percent over 100.
M. 6.3.14: Identify a proportion from given information.
M. 6.3.13: Calculate a proportion for missing information.
M. 6.3.10: Create a proportion or ratio from a given word problem.

Alabama Alternate Achievement Standards

AAS Standard: M.G.AAS.10.36 Use geometric shapes to describe real-world objects.