Theorems about triangles and quadrilaterals
A carpenter has just built a wooden gate frame. She wants to be sure every corner is a perfect right angle, but she left her framing square at home. All she has is a tape measure. Can she check the corners by measuring just two lengths? By the end of this page you will be able to prove that she can, and say exactly why.
Interior and exterior angles
Every triangle you will ever draw has angles that add up to the same number. Here is why.
Draw a line through vertex $C$ parallel to $\overline{AB}$. The parallel line makes angle 1 equal to $\angle A$ and angle 3 equal to $\angle B$, because they are alternate interior angles. Angles 1, 2 and 3 together make a straight angle.
An exterior angle is formed when you extend one side of the triangle. It sits outside the triangle, next to one interior angle. The other two interior angles are called its remote interior angles.
Why does this work? The exterior angle and the interior angle next to it form a linear pair, so together they make $180^\circ$. The three interior angles also make $180^\circ$. Both pairs are "$180^\circ$ minus the same angle", so the exterior angle must equal the other two interior angles added together. In the figure, $58^\circ + 67^\circ = 125^\circ$.
A carpenter is building a wooden roof truss in the shape of a triangle. The first base angle measures $42^\circ$ and the second base angle measures $55^\circ$. Calculate…
A carpenter is building a wooden roof truss in the shape of a triangle. The first base angle measures $42^\circ$ and the second base angle measures $55^\circ$. Calculate the measure of the third angle at the peak of the truss.
Isosceles triangles
Pennants, roof trusses and bridge supports are often isosceles. The two equal sides are the legs, the third side is the base, and the angle between the legs is the vertex angle. The other two angles are the base angles.
Why is it true? Draw the segment from $P$ to the midpoint $M$ of $\overline{QR}$. The triangles $\triangle PQM$ and $\triangle PRM$ share $\overline{PM}$, have $\overline{PQ} \cong \overline{PR}$, and have $\overline{QM} \cong \overline{RM}$. They are congruent by SSS, so $\angle Q \cong \angle R$ by CPCTC. An equilateral triangle has three congruent sides, so all three angles are congruent. Each one is $180^\circ \div 3 = 60^\circ$.
A local high school sports team uses an isosceles triangle pennant. The angle at the vertex (the tip of the pennant) is $34^\circ$. Since the top and bottom edges of the…
A local high school sports team uses an isosceles triangle pennant. The angle at the vertex (the tip of the pennant) is $34^\circ$. Since the top and bottom edges of the pennant are equal in length, determine the measure of the two base angles attached to the flagpole.
A bridge support is built as an isosceles triangle with two equal legs. The base angles are represented by the expressions $(2x + 10)^\circ$ and $(4x - 20)^\circ$. Set…
A bridge support is built as an isosceles triangle with two equal legs. The base angles are represented by the expressions $(2x + 10)^\circ$ and $(4x - 20)^\circ$. Set up an equation to find the value of $x$, and then determine the measure of the vertex angle $y$ at the top of the support.
An aircraft wing flap extends outward, creating an exterior angle with the main wing structure. The triangular cross-section of the flap is an isosceles triangle where…
An aircraft wing flap extends outward, creating an exterior angle with the main wing structure. The triangular cross-section of the flap is an isosceles triangle where the two base angles are equal. If the exterior angle adjacent to the vertex angle measures $116^\circ$, find the measure of the base angles.
Midsegments of a triangle
A midsegment of a triangle connects the midpoints of two sides. A triangle has three of them. Look at what the midsegment does to the triangle.
Why does the half show up? The small triangle $\triangle ADE$ is a scaled-down copy of $\triangle ABC$ (they are similar by SAS with a scale factor of $\tfrac{1}{2}$). A scale factor of $\tfrac12$ makes every length half as big, and it makes $\overline{DE}$ parallel to $\overline{BC}$.
You can also prove it with coordinates. Put $B$ at the origin and $C$ on the $x$-axis. Take $A(4, 6)$, $B(0, 0)$ and $C(8, 0)$. The midpoint of $\overline{AB}$ is $D(2, 3)$ and the midpoint of $\overline{AC}$ is $E(6, 3)$. Both $\overline{DE}$ and $\overline{BC}$ have slope $0$, so they are parallel. Also $DE = 6 - 2 = 4$ and $BC = 8$, so $DE = \tfrac12 BC$. For a general proof, use letters instead of numbers: $B(0, 0)$, $C(2c, 0)$, $A(2p, 2q)$. Doubling the coordinates keeps the midpoints free of fractions.
A hiking trail forms a large triangle between three base camps. A new shortcut is built connecting the exact midpoints of two sides of the trail network. If the third…
A hiking trail forms a large triangle between three base camps. A new shortcut is built connecting the exact midpoints of two sides of the trail network. If the third side of the trail network is $4.8 \text{ miles}$ long, determine the length of the new shortcut using the Triangle Midsegment Theorem.
A landscape architect is designing a triangular garden. The outer boundary has side lengths of $32 \text{ ft}$, $40 \text{ ft}$, and $46 \text{ ft}$. A smaller…
A landscape architect is designing a triangular garden. The outer boundary has side lengths of $32 \text{ ft}$, $40 \text{ ft}$, and $46 \text{ ft}$. A smaller triangular flower bed is created by connecting the exact midpoints of the outer boundary. Calculate the perimeter of the smaller flower bed and state the geometric theorem used to find it.
A graphic design software uses the Triangle Midsegment Theorem to scale objects. Consider $\triangle ABC$ with $D$ as the midpoint of $\overline{AB}$ and $E$ as the…
A graphic design software uses the Triangle Midsegment Theorem to scale objects. Consider $\triangle ABC$ with $D$ as the midpoint of $\overline{AB}$ and $E$ as the midpoint of $\overline{AC}$. Construct a logical argument (or a two-column proof) to prove that the midsegment $\overline{DE}$ is parallel to $\overline{BC}$ and $DE = \frac{1}{2}BC$. You may use coordinate geometry or similar triangles in your reasoning.
Medians and the centroid
A median of a triangle is a segment from a vertex to the midpoint of the opposite side. Notice the difference from a midsegment: a median starts at a vertex, a midsegment connects two midpoints. All three medians cross at one point, the centroid.
The centroid is the balancing point of the triangle. If you cut a triangle out of cardboard, it will balance on a pencil tip placed at $G$.
You can see the $2:1$ split with coordinates. Take $A(0, 0)$, $B(6, 0)$ and $C(0, 6)$. The centroid is the average of the vertices: $G = \left(\tfrac{0+6+0}{3}, \tfrac{0+0+6}{3}\right) = (2, 2)$. The median from $C$ goes to $M(3, 0)$, the midpoint of $\overline{AB}$. Then $CG = \sqrt{2^2 + 4^2} = 2\sqrt5$ and $CM = \sqrt{3^2 + 6^2} = 3\sqrt5$, so $CG = \tfrac23 CM$. ✓
Two facts about area will help you with proofs later.
A triangular metal plate is designed for an aerospace component. The balancing point, or centroid, lies on a median drawn from vertex $A$ to the midpoint of the opposite…
A triangular metal plate is designed for an aerospace component. The balancing point, or centroid, lies on a median drawn from vertex $A$ to the midpoint of the opposite side. If the total length of this median is $18 \text{ inches}$, find the distance from vertex $A$ to the centroid.
A drone's flight path maps out a triangle over a national park. The centroid $G$ marks the location of a communication tower. The median from vertex $B$ passes through…
A drone's flight path maps out a triangle over a national park. The centroid $G$ marks the location of a communication tower. The median from vertex $B$ passes through $G$ to point $M$ on the opposite side. The distance $BG$ is given as $3z - 4 \text{ miles}$ and the distance $GM$ is $z + 5 \text{ miles}$. Find the total length of the median $BM$.
A surveyor divides a large triangular plot of land into six smaller regions by drawing all three medians, which intersect at the centroid. Prove that all six smaller…
A surveyor divides a large triangular plot of land into six smaller regions by drawing all three medians, which intersect at the centroid. Prove that all six smaller triangular regions have exactly the same area, ensuring the land is divided equally among six heirs.
Properties of parallelograms
A parallelogram is a quadrilateral with both pairs of opposite sides parallel. Windows, scissor lifts and parking spaces are all parallelograms. Everything below follows from this one definition.
Where do these come from? Consecutive angles are same-side interior angles between parallel lines, so they add to $180^\circ$. For opposite sides, draw a diagonal. The two triangles it makes are congruent by ASA, because the parallel sides give alternate interior angles and the diagonal is shared. Then CPCTC gives congruent opposite sides and opposite angles. Another pair of congruent triangles, formed by the two diagonals, shows that the diagonals bisect each other.
An architect designs a custom window in the shape of a parallelogram. If one of the acute interior angles measures $68^\circ$, find the measure of the consecutive obtuse…
An architect designs a custom window in the shape of a parallelogram. If one of the acute interior angles measures $68^\circ$, find the measure of the consecutive obtuse interior angle. State the theorem that justifies your answer.
An industrial scissor lift uses intersecting steel beams that form a parallelogram. The point where the two diagonal beams intersect bisects each beam. If one entire…
An industrial scissor lift uses intersecting steel beams that form a parallelogram. The point where the two diagonal beams intersect bisects each beam. If one entire diagonal beam is $14.5 \text{ feet}$ long, what is the distance from the corner of the parallelogram to the intersection point?
A city planner is striping a new angled parking lot where each space is a parallelogram. The opposite sides of one parking space have lengths represented by…
A city planner is striping a new angled parking lot where each space is a parallelogram. The opposite sides of one parking space have lengths represented by $5w - 12 \text{ ft}$ and $2w + 15 \text{ ft}$. Determine the value of $w$ and the actual length of these parallel sides.
Proving a quadrilateral is a parallelogram
The parallelogram theorems go one way: "If it is a parallelogram, then...". To prove that a shape is a parallelogram you need the converses, which are the tests below.
The parallelogram theorems are tools for using a parallelogram. The five tests above are tools for proving that you have one. When you see "prove that $ABCD$ is a parallelogram", pick whichever test fits the information you are given. These reasons are also useful: the converse of the Alternate Interior Angles Theorem, and "if same-side interior angles are supplementary, the lines are parallel."
A proof shows that a statement is always true. To show that a statement is false, one counterexample is enough. It must satisfy every condition in the claim and still break the conclusion.
A student drafting a floor plan claims: "If a quadrilateral has one pair of parallel sides and one pair of congruent sides, it must be a parallelogram." Prove or…
A student drafting a floor plan claims: "If a quadrilateral has one pair of parallel sides and one pair of congruent sides, it must be a parallelogram." Prove or disprove this statement. If it is false, name the specific type of quadrilateral that serves as a counterexample.
In a mechanical linkage, a quadrilateral framework is designed such that both pairs of opposite interior angles are congruent. Write a formal proof showing that this…
In a mechanical linkage, a quadrilateral framework is designed such that both pairs of opposite interior angles are congruent. Write a formal proof showing that this condition alone is sufficient to prove the framework is a parallelogram. (Hint: Use the fact that the sum of interior angles is $360^\circ$).
Rectangles and rhombuses
Two special parallelograms get their own names. A rectangle is a parallelogram with four right angles. A rhombus is a parallelogram with four congruent sides. A square is both.
Why does one right angle do it? In a parallelogram, consecutive angles add to $180^\circ$. A right angle forces its neighbors to be $90^\circ$, and the opposite angle is also $90^\circ$. For the rhombus, each diagonal cuts the shape into two isosceles triangles, and that is what makes the diagonals bisect the angles.
A contractor is installing a rectangular foundation for a new house. To verify the corners are perfect right angles, she measures the diagonals. The first diagonal…
A contractor is installing a rectangular foundation for a new house. To verify the corners are perfect right angles, she measures the diagonals. The first diagonal measures $5x + 12 \text{ feet}$ and the second diagonal measures $7x - 4 \text{ feet}$. Calculate the value of $x$ that ensures the foundation is a true rectangle.
A jeweler is cutting a gemstone into a rhombus shape. The diagonals of the rhombus intersect at an angle labeled $(3y + 15)^\circ$. Since the diagonals of a rhombus must…
A jeweler is cutting a gemstone into a rhombus shape. The diagonals of the rhombus intersect at an angle labeled $(3y + 15)^\circ$. Since the diagonals of a rhombus must be perpendicular, solve for the value of $y$.
A carpenter builds a wooden gate frame. He ensures that the opposite sides are equal in length, which guarantees the frame is a parallelogram. He then measures the two…
A carpenter builds a wooden gate frame. He ensures that the opposite sides are equal in length, which guarantees the frame is a parallelogram. He then measures the two diagonals and finds they are exactly the same length. Write a short explanation of why having equal diagonals in a parallelogram proves that the gate frame is a true rectangle.
Rhombus perimeter, kites and the quadrilateral family
The diagonals of a rhombus cross at right angles and bisect each other. That cuts it into four congruent right triangles. The half-diagonals are the legs and the side of the rhombus is the hypotenuse.
A kite is a quadrilateral with two pairs of adjacent congruent sides, where opposite sides are not congruent. Think of a kite flying in the sky.
Useful tools for proofs: the Perpendicular Bisector Theorem says that every point on the perpendicular bisector of a segment is the same distance from its two endpoints. So if one diagonal is the perpendicular bisector of the other, each endpoint of the other diagonal is equidistant from both ends of the first. That makes two pairs of adjacent congruent sides, which is a kite. Also remember that a quadrilateral whose diagonals bisect each other is a parallelogram. A kite whose diagonals bisect each other would therefore be a parallelogram with perpendicular diagonals, which means a rhombus.
Here is how all these shapes fit together.
Going down a line adds a property. A square is a rectangle (four right angles) and a rhombus (four congruent sides), so it has every property of both.
A custom stained glass window features a large central rhombus. The stained glass artist measures the diagonals of the rhombus to be $18 \text{ inches}$ and…
A custom stained glass window features a large central rhombus. The stained glass artist measures the diagonals of the rhombus to be $18 \text{ inches}$ and $24 \text{ inches}$. Using the properties of a rhombus, calculate the perimeter of this central glass piece.
A jewelry designer creates a pendant in the shape of a quadrilateral. The design specifies that the diagonals are perpendicular to each other, but only one diagonal…
A jewelry designer creates a pendant in the shape of a quadrilateral. The design specifies that the diagonals are perpendicular to each other, but only one diagonal bisects the other. Prove that this pendant is a kite and cannot be a rhombus.
A civil engineer is designing a city plaza in the shape of a quadrilateral. She states: "If the diagonals of a quadrilateral are perpendicular to each other and bisect…
A civil engineer is designing a city plaza in the shape of a quadrilateral. She states: "If the diagonals of a quadrilateral are perpendicular to each other and bisect each other, the plaza must be a perfect square." Prove or disprove this statement. If false, identify the correct shape and explain what additional property is needed to guarantee a square.
Putting a proof together
Proofs feel hard because you must choose your own path. Use the same routine every time.
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Draw the figure and mark everything that is Given.
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Write "Given" and "Prove" in your own words.
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Plan backwards: which theorem or test gives you the conclusion? What would you need to know first?
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Add any helper you need, usually a diagonal or a midpoint segment.
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Write each statement with its reason: Given, a definition, a theorem, a postulate or an earlier step.
Here is a model proof that shows the format.
Given: $ABCD$ is a rectangle. Prove: $\overline{AC} \cong \overline{BD}$.
| Statement | Reason |
|---|---|
| 1. $ABCD$ is a rectangle | Given |
| 2. $\overline{AB} \cong \overline{DC}$ | Opposite sides of a parallelogram are congruent (a rectangle is a parallelogram) |
| 3. $\angle ABC \cong \angle DCB$ | All angles of a rectangle are right angles, so they are congruent |
| 4. $\overline{BC} \cong \overline{CB}$ | Reflexive Property |
| 5. $\triangle ABC \cong \triangle DCB$ | SAS (steps 2, 3, 4) |
| 6. $\overline{AC} \cong \overline{DB}$ | CPCTC |
For the proof exercises below, here are some chains that often work:
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Parallel lines give congruent angles, such as corresponding or alternate interior angles.
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Congruent angles in a triangle give congruent sides (the converse of the Isosceles Triangle Theorem).
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A midsegment gives a parallel segment and half a length.
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For the inner shape of a rhombus, look at the diagonals of the rhombus. The sides of the inner shape are parallel to those diagonals, and the diagonals of a rhombus are perpendicular.
A solar panel mount is shaped like an isosceles triangle $\triangle PQR$ with $\overline{PQ} \cong \overline{PR}$. A horizontal support beam $\overline{ST}$ is installed…
A solar panel mount is shaped like an isosceles triangle $\triangle PQR$ with $\overline{PQ} \cong \overline{PR}$. A horizontal support beam $\overline{ST}$ is installed such that it is parallel to the base $\overline{QR}$, intersecting $\overline{PQ}$ at $S$ and $\overline{PR}$ at $T$. Prove that the newly formed triangle $\triangle PST$ is also isosceles.
A decorative quilt features a large, irregular quadrilateral. The quilter connects the midpoints of all four consecutive sides of this quadrilateral to form a smaller…
A decorative quilt features a large, irregular quadrilateral. The quilter connects the midpoints of all four consecutive sides of this quadrilateral to form a smaller inner shape. Prove that this inner shape is always a parallelogram, regardless of the original quadrilateral's dimensions. (Hint: Draw a diagonal of the large quadrilateral and apply the Triangle Midsegment Theorem).
An architectural framing technique involves taking a rhombus-shaped window and connecting the midpoints of its four sides to create an inner frame. Prove that the…
An architectural framing technique involves taking a rhombus-shaped window and connecting the midpoints of its four sides to create an inner frame. Prove that the quadrilateral formed by connecting the midpoints of a rhombus is always a rectangle.
Back to the start: the carpenter measures the sides to be sure the frame is a parallelogram (opposite sides equal). Then she measures the two diagonals. If they are congruent, the frame is a rectangle: the triangles formed by two sides and a diagonal are congruent by SSS, and their matching angles are supplementary parallelogram angles, which forces them all to be $90^\circ$.