Triangle Truths Proof Guide Geometry Reference Guide Amplify Math • Triangle Truths
Midsegment Triangle Congruence
Quick, student-friendly proof showing all four triangles are congruent.
The Problem
Points \(D\), \(E\), and \(F\) are midpoints of \(\overline{AB}\), \(\overline{AC}\), and \(\overline{BC}\).
Prove: \(\triangle ADE \cong \triangle DBF \cong \triangle EFC \cong \triangle FED\).
A C B D E F
Student Answer (Ready to Type In)
Full Credit Proof
By the Triangle Midsegment Theorem , each segment connecting two midpoints is equal to half the length of the opposite side:
\(DE = \frac{1}{2} BC = BF = FC\)
\(DF = \frac{1}{2} AC = AE = EC\)
\(EF = \frac{1}{2} AB = AD = DB\)
Because of this, all four triangles (\(\triangle ADE\), \(\triangle DBF\), \(\triangle EFC\), and \(\triangle FED\)) have sides of lengths \(\frac{1}{2}AB\), \(\frac{1}{2}AC\), and \(\frac{1}{2}BC\). Therefore, they are all congruent by SSS (Side-Side-Side) !
Step-by-Step Breakdown
Step 1
Split by Midpoints
\(D, E, F\) cut each side into two equal halves:
AD = DB = ½ AB
Step 2
Midsegment Theorem
Inside segments are half the outer sides:
DE = ½ BC
Step 3
Match Sides (SSS)
Every triangle has identical side lengths:
✓ 3 matching sides
Side Length Matching Table
Triangle
Side 1 (\(= \frac{1}{2}AB\))
Side 2 (\(= \frac{1}{2}AC\))
Side 3 (\(= \frac{1}{2}BC\))
\(\triangle ADE\)
\(AD\)
\(AE\)
\(DE\)
\(\triangle DBF\)
\(DB\)
\(DF\)
\(BF\)
\(\triangle EFC\)
\(EF\)
\(EC\)
\(FC\)
\(\triangle FED\) (center)
\(FE\)
\(DF\)
\(ED\)
Key Terms to Include: Mentioning the Midsegment Theorem and SSS Congruence is what teachers look for when grading!
Side Splitter Synthesis Guide Geometry Synthesis Sheet Amplify Math • Screen 12 of 14
Proportional Side Relationships
Key geometric truths when a line segment divides two triangle sides proportionally.
Synthesis Question
Given \(\frac{AD}{AB} = \frac{AE}{AC} = 0.5\):
"What are some relationships that exist in triangles where a line segment divides two sides proportionally?"
A B C D E
Short Student Answers (Pick Any Option)
Classroom Ready
Option 1: Quick Bullet Points (Recommended)
• The segment is parallel to the base: \(\overline{DE} \parallel \overline{BC}\).similar to the large triangle: \(\triangle ADE \sim \triangle ABC\).corresponding angles are equal : \(\angle ADE = \angle B\) and \(\angle AED = \angle C\).
Option 2: Just 2 Short Sentences
"The dividing line segment is parallel to the base (\(\overline{DE} \parallel \overline{BC}\)). Because of this, the small triangle is similar to the big triangle (\(\triangle ADE \sim \triangle ABC\)), so all corresponding angles are equal and \(\frac{DE}{BC} = 0.5\)."
The 4 Big Relationships Explained
1
Parallel Lines
By the Converse of the Side-Splitter Theorem , if two sides are divided proportionally, the dividing segment is parallel to the third side:
\(\overline{DE} \parallel \overline{BC}\)
2
Similar Triangles
The top triangle and the whole triangle share angle \(A\) and have proportional sides, making them similar by SAS:
\(\triangle ADE \sim \triangle ABC\)
3
Equal Corresponding Angles
Because the lines are parallel cut by transversals, corresponding angles are equal:
\(\angle ADE \cong \angle B\) & \(\angle AED \cong \angle C\)
4
Same Scale Factor
The base ratio matches the side ratio. If sides are scaled by \(0.5\), the base is too:
\(\frac{DE}{BC} = 0.5 \implies DE = \frac{1}{2} BC\)
Teacher Tip: The two most important buzzwords teachers grade for here are parallel (\(DE \parallel BC\)) and similar triangles (\(\triangle ADE \sim \triangle ABC\))!
Proportional Triangles Check Sheet Assessment Answer Key Amplify Math • Screen 13 of 14
Show What You Know: Proportions
Quick selection guide for triangle relationships when sides are divided proportionally.
The Exact Checkbox Answers to Select
Select Exactly 2
✓
\(\overline{JK}\) is parallel to \(\overline{YZ}\).
TRUE — By the Converse of the Side-Splitter Theorem, any proportional divider is parallel to the base.
✓
\(\frac{XY}{XJ} = \frac{YZ}{JK}\)
TRUE — Both are the ratio of \(\frac{\text{big triangle side}}{\text{small triangle side}}\) from similar triangles \(\triangle XYZ \sim \triangle XJK\).
Full Statement Breakdown & Why
Statement 1
\(\overline{JK}\) is parallel to \(\overline{YZ}\).
Whenever a line divides two sides proportionally, it is guaranteed to be parallel to the 3rd side.
CHECK (TRUE)
Statement 2
For \(\overline{JK}\) to be parallel, points \(J\) and \(K\) must be midpoints.
False: Any ratio works (like \(1:3\) or \(2:5\)). They do not have to be midpoints!
DO NOT CHECK
Statement 3
\(\frac{XY}{XJ} = \frac{XK}{XZ}\)
False: Upside-down fraction! Left is \(\frac{\text{big}}{\text{small}}\), but right is \(\frac{\text{small}}{\text{big}}\). Should be \(\frac{XZ}{XK}\).
DO NOT CHECK
Statement 4
\(\frac{XY}{XJ} = \frac{YZ}{JK}\)
Both sides compare \(\frac{\text{big triangle}}{\text{small triangle}}\) for corresponding sides. This is the definition of similarity.
CHECK (TRUE)
Statement 5
\(\frac{XY}{XJ} = \frac{XZ}{JK}\)
False: Mismatched sides! \(XZ\) corresponds to \(XK\), not the base \(JK\).
DO NOT CHECK
Remember the similarity order: \(\triangle XJK \sim \triangle XYZ\). So \(\frac{XY}{XJ} = \frac{XZ}{XK} = \frac{YZ}{JK}\).
