Given: modifying above upper S upper D with bar perpendicular to modifying above upper H upper T with bar; modifying above upper S upper H with bar congruent to modifying above upper S upper T with bar
Prove:triangleSHD = triangleSTD
Triangle HST is split into two triangles by vertical line SD.A triangle is split into two triangles by vertical line SD. The vertex at the bottom left is labeled H. The vertex at the bottom right is labeled T. The vertex at the top in the middle is labeled S. The vertical line down the middle of the triangle starts at point S and ends on line segment HT at a point labeled D. Line segments HS and ST are marked as congruent.
The image of a two column proof is shown. The left column is titled Statement, and the right column is titled Reason.
Statement Reason
1. Triangle HST is Given
split into two
triangles by line SD
2. HS = ST Given
3. SDSH is a definition of midpoint
parallelogram
4. SAS triangle SAS postulate
congruence (SS)
SDS ~ TDS
5. Triangle SHD ~ CPCTC
triangle STD
6. Angle SHD = definition of congruent
angle STD
7. Triangle SHD = SAS (SSA) congruence
triangle STD theorem
Statement Reason
1. Segments HS and ST are congruent Given
2. Line SD splits triangle HST into two triangles Given
3. Segment SD is perpendicular to segment HT Given
4. Triangle SHD and triangle STD share segments
SD, SH, and ST Common Side
5. Segment SD is congruent to itself Reflexive Property
6. Segment SH is congruent to segment ST Transitive Property (from 1 and 5)
7. Segment HD is congruent to segment TD Transitive Property (from 1 and 6)
8. Angle SHD is congruent to angle STD Perpendicular lines form congruent adjacent angles
9. Triangle SHD and triangle STD have three pairs
of congruent corresponding parts CPCTC (Corresponding Parts of Congruent Triangles are Congruent)
To prove: triangle SHD = triangle STD
Statement | Reason
----------|----------
1. Triangle HST is split into two triangles by vertical line SD | Given
2. Triangle HST is split into triangles SHD and STD | Definition of triangle splitting
3. Line segment HS is congruent to line segment ST | Given
4. Line segment HD is congruent to line segment TD | Vertical line SD divides triangle HST into congruent parts, so corresponding parts are congruent
5. Line segment SH is congruent to line segment ST, and line segment HD is congruent to line segment TD | Consequence of congruence property
6. Triangle SHD is congruent to triangle STD | Side-Side-Side (SSS) congruence criterion
7. Therefore, triangle SHD = triangle STD | Two congruent triangles are equal
To summarize, we can prove that triangle SHD is equal to triangle STD by using the given information and applying the congruence properties of the sides and segments within the triangles.