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Phew. I spent two days finishing Unit 3, Congruence, from the High
School Geometry course on Khan Academy. I went through a lot of geometry
proofs. I can definitely feel my progress slowing down a little, but it
was worth the time.
When two triangles have the same lengths for all corresponding sides and
the same measures for all corresponding angles, we can say the two
triangles must be congruent. However, we do not necessarily need all of
those facts to prove congruence. There are several shortcuts: SSS, SAS,
ASA, AAS, and HL.
SSS stands for side-side-side. When three pairs of corresponding sides
are congruent, the two triangles are congruent.
Proof: For triangles ABC and DEF, suppose we are given that segment AB
is congruent to segment DE, segment BC is congruent to segment EF, and
segment AC is congruent to segment DF. We map segment AB onto segment DE
through a rigid transformation. Rigid transformations preserve distances
and angle measures. Then draw a circle centered at D with radius DF, and
another circle centered at E with radius EF. The image of point C must
lie at one of the intersections of the two circles. If the image point
C' lands on F, we are done. Otherwise, it lies on the opposite side of
segment DE, and reflecting the triangle across DE maps C' onto F.
SAS stands for side-angle-side. If two corresponding sides are
congruent, and the included angles between them are congruent, then the
two triangles are congruent.
Proof: For triangles ABC and DEF, suppose we are given that segment AB
is congruent to segment DE, segment BC is congruent to segment EF, and
angle B is congruent to angle E. We map segment AB onto segment DE
through a rigid transformation. Rigid transformations preserve distances
and angle measures. Then draw a circle centered at E with radius EF. The
image point C' lies at the intersection of the circle and the ray
forming angle E with segment DE. If C' lands on F, we are done.
Otherwise, it lies on the opposite side of segment DE, and reflecting
the triangle across DE maps C' onto F.
ASA stands for angle-side-angle. When two pairs of corresponding angles
are congruent, and the included side between them is congruent, then the
two triangles are congruent.
Proof: For triangles ABC and DEF, suppose we are given that segment BC
is congruent to segment EF, angle B is congruent to angle E, and angle C
is congruent to angle F. We map segment BC onto segment EF through a
rigid transformation. Rigid transformations preserve distances and angle
measures. The two angles determine two rays, and their intersection
determines the image point A'. If A' lands on D, we are done. Otherwise,
it lies on the opposite side of segment EF, and reflecting the triangle
across EF maps A' onto D.
AAS stands for angle-angle-side. When two pairs of corresponding angles
are congruent, and a non-included side is congruent, then the two
triangles are congruent.
Proof: The interior angles of a triangle sum to 180°. Since two angles
are already known, we can calculate the third angle. Then we can apply
the ASA postulate to prove the triangles are congruent.
HL stands for hypotenuse-leg. If the hypotenuses and one pair of
corresponding legs of two right triangles are congruent, then the
triangles are congruent.
Proof: This is a special case of right triangles. Using the Pythagorean
Theorem, we can calculate the missing side length. Once we know all
three side lengths, we can apply the SSS postulate.
There is also an SSA case—side-side-angle—which is not a valid triangle
congruence postulate.
Suppose we are given that segment AB is congruent to segment DE, segment
AC is congruent to segment DF, and angle C is congruent to angle F. We
map segment AC onto segment DF through a rigid transformation. Rigid
transformations preserve distances and angle measures. Then draw a
circle centered at D with radius DE. The image point B' may lie at one
of two intersections between the circle and the ray forming angle F with
segment DF. In some cases, both positions satisfy the given conditions
but form different triangles. Therefore, SSA is ambiguous and does not
guarantee triangle congruence.
The unit also covers the reflexive, symmetric, and transitive properties
of congruence.
Reflexive property: A relation is reflexive if every object is related
to itself. For example, if triangles ABC and ADC share side AC, then
segment AC is congruent to itself by the reflexive property.
Symmetric property: If A is related to B, then B is also related to A.
For example, if segment AB is congruent to segment BC, then segment BC
is also congruent to AB by the symmetric property.
Transitive property: If A is related to B, and B is related to C, then A
is related to C. For example, if angle A is congruent to angle B, and
angle B is congruent to angle C, then angle A is congruent to angle C by
the transitive property.
In the evening, I spent some time watching the keynote of Google I/O
2026. Google introduced a new family of Gemini models—Gemini 3.5 Flash,
which is highly capable and outperforms Gemini 3.1 Pro. It even
outperforms Claude Opus 4.7 and GPT 5.5 in some benchmarks. The crazy
thing is that its output speed is incredibly fast, reaching nearly 280
tokens per second. The Pro version will be released later next month.
Google also lowered the price of the highest-tier Gemini subscription
from $250 to $200. One thing that stops me from using Gemini is that I
cannot opt out of AI traning unless I gave up my chat history. I feel
uncomfortable about that, almost as if I am being force into it.
During the keynote, Google—an AI-first company—also demonstrated how
deeply they are integrating Gemini 3.5 Flash into their products and
across the entire ecosystem: Google Search, Shopping, YouTube, Gmail,
and more. In the near future, Google Search will leverage Antigravity to
generate interactive demos from scratch to help people understand new
ideas or concepts. That was probably the feature I liked the most.
I feel deeply touched by how AI may completely change people's work and
daily lives. Software may become dispoable and one-time use. At the end
of the keynote, Google introduced a new version of Google Glasses. I
guess I probably will not be able to experience them, but I can already
think of one use case that would suit me very well right now: whenever I
need help with a math problem, I would not need to take out my phone and
snap a photo anymore. I could simply talk to Gemini directly through the
glasses—it sees what I see.
Oh, and there was also another keynote dedicated specifically to
developers. I guess I need to set aside some extra time to watch that as
well.
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